GENERALIZED PUSHOVER ANALYSIS FOR UNSYMMETRICAL …

GENERALIZED PUSHOVER ANALYSIS FOR UNSYMMETRICAL-PLAN BUILDINGS

A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OFMIDDLE EAST TECHNICAL UNIVERSITY

BY

KAAN KAATSIZ

IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR

THE DEGREE OF MASTER OF SCIENCEIN

CIVIL ENGINEERING

JULY 2012

Approval of the thesis:


submitted by KAAN KAATSIZ in partial fulfillment of the requirements for the degree ofMaster of Science in Civil Engineering Department, Middle East Technical Universityby,

Prof. Dr. Canan ÖzgenDean, Graduate School of Natural and Applied Sciences

Prof. Dr. Güney ÖzcebeHead of Department, Civil Engineering

Prof. Dr. Haluk SucuogluSupervisor, Civil Engineering Department, METU

Examining Committee Members:

Prof. Dr. Polat GülkanCivil Engineering Dept., Çankaya University

Prof. Dr. Haluk SucuogluCivil Engineering Dept., METU

Assoc. Prof. Dr. Murat Altug ErberikCivil Engineering Dept., METU

Assoc. Prof. Dr. Afsin SarıtasCivil Engineering Dept., METU

Joseph Kubin, M.Sc.Civil Engineer, PROTA

Date: 17.07.2012

I hereby declare that all information in this document has been obtained and presentedin accordance with academic rules and ethical conduct. I also declare that, as requiredby these rules and conduct, I have fully cited and referenced all material and results thatare not original to this work.

Name, Last Name: KAAN KAATSIZ

Signature :

iii

ABSTRACT


Kaatsız, Kaan

M.S., Department of Civil Engineering

Supervisor : Prof. Dr. Haluk Sucuoglu

July 2012, 104 pages

Nonlinear response history analysis is regarded as the most accurate analysis procedure for

estimating seismic response. Approximate analysis procedures are also available for the deter-

mination of seismic response and they are preferred over nonlinear response history analysis

since much less computational effort is required and good response prediction is achieved by

employing rather simple concepts.

A generalized pushover analysis procedure is developed in this thesis study as an approxi-

mate analysis tool for estimating the inelastic seismic response of structures under earthquake

ground excitations. The procedure consists of applying generalized force vectors to the st-

ructure in an incremental form until a prescribed target interstory drift demand is achieved.

Corresponding generalized force vectors are derived according to this target drift parameter

and include the contribution of all modes. Unlike many approximate analysis procedures, res-

ponse of the structure is directly obtained from generalized pushover analysis results without

employing a modal combination rule, eliminating the errors cultivating from these methods.

Compared to nonlinear response history analysis, generalized pushover analysis is less de-

manding in computational effort and its implementation is simpler relative to other approxi-

mate analysis procedures. It is observed that the proposed analysis procedure yields results

iv

accurately in comparison to the other nonlinear pushover analysis methods. Accordingly it

can be suggested as a convenient and sound analysis tool.

Keywords: Structural evaluation, multi modal pushover analysis, generalized force vectors

v

ÖZ

ÜÇ BOYUTLU BURULMALI SISTEMLER IÇIN GENEL ITME ANALIZI

Kaatsız, Kaan

Yüksek Lisans, Insaat Mühendisligi Bölümü

Tez Yöneticisi : Prof. Dr. Haluk Sucuoglu

Temmuz 2012, 104 sayfa

Yapıların sismik tepkilerin belirlenmesinde, dogrusal olmayan dinamik analiz en kesin sonuç

veren analiz yöntemi olarak kabul görmektedir. Bu yöntemin yanında, sismik tepkilerin tah-

min edilebilmesi için yaklasık analiz yöntemleri de mevcuttur. Yaklasık analiz yöntemleri ile

nispeten basit kavramlar kullanılarak sismik yapı davranısı yeterli dogrulukta kestirilebilmek-

tedir. Ayrıca bu yöntemler çok daha az islem yükü gerektirdigi ve uygulama basitligine sahip

oldukları için pratikte dogrusal olmayan dinamik analiz yöntemi yerine tercih edilmektedirler.

Bu tez çalısmasında yapıların deprem yer hareketleri altında dogrusal olmayan sismik tepkile-

rini tahmin edebilmek için genel itme analizi yöntemi gelistirilmistir. Yaklasık bir analiz yön-

temi olan genel itme analizi, dogrusal olmayan statik itme analizi prensibini kullanmaktadır.

Bu yöntemde, mod birlestirme kuralları kullanarak hesaplanan genel kuvvet vektörleri yapıya

artımsal olarak bir katın önceden belirlenmis kat arası öteleme talebine ulasıncaya kadar uy-

gulanmaktadır. Bir kat ile ilgili olarak hesaplanan genel yük vektörleri, bir kat arası öteleme

parametresine göre türetilir ve tüm modların katkısını içerir. Pek çok yaklasık analiz yöntemi-

nin aksine, yapının tepkisi direk olarak genel itme analizi sonuçlarından elde edilmektedir ve

ilaveten herhangi bir mod birlestirme kuralının uygulanmasına ihtiyaç duymamaktadır. Bu sa-

yede, dogrusal olmayan tepki parametrelerinin mod birlestirme kuralı ile birlestirilmesinden

vi

kaynaklanan hatalar ortadan kaldırılmaktadır. Dogrusal olmayan dinamik analiz ile karsılastı-

rıldıgında, genel itme analizinin islemsel güç gereksinimi çok daha azdır ve diger çok modlu

statik itme analizi yöntemlerine göre uygulanması daha basittir. Önerilen analiz yöntemi diger

çok modlu dogrusal olmayan itme analizi yöntemleri ile kıyaslandıgında daha dogru sonuç-

lar üretmektedir. Bu sebeplerle, genel itme analizi kullanıslı ve güvenilir bir analiz yöntemi

olarak önerilmektedir.

Anahtar Kelimeler: Yapısal degerlendirme, çok modlu statik itme analizi, genel yük vektörleri

vii

To my mother...

viii

ACKNOWLEDGMENTS

This study was conducted under the supervision of Prof. Dr. Haluk Sucuoglu. I would like

to express my earnest thanks and appreciations for his support, guidance, encouragement and

criticisms during this study. It was a great honor and pleasure to work with him.

I would like to thank to my family for their endless support and love. During my study,

I felt their encouragement and guidance all the time. Their enthusiasm about this study was

always my main source of motivation. The patience and support shown by them are thankfully

acknowledged.

I also want to extend my thanks to my office mates M. Basar Mutlu, F. Soner Alıcı, M. Can

Yücel, Ahmet Kusyılmaz, Sadun Tanıser and Alper Ö. Gür. The joyful times that we shared

in Room Z01 and their enormous support will always be remembered with pleasure.

Sincere thanks to M. Basar Mutlu and F. Soner Alıcı for their great friendship. Long, tiresome

yet productive and cheerful study sessions we had together over the years, their assistance in

hard times and great memories that we shared together are gladly remembered.

Finally, my very special thanks go to my mother, who passed away a short time ago before the

completion of this study. Her great love and guidance have been essential for me throughout

my life. She always put me before herself and even during her illness she wanted me to

concentrate on my work. Her self-devotion, love, encouragement, support and guidance are

greatly acknowledged. Even though she deceased at the final phase of the study, I am sure that

she sees what her son achieved and she is proud of me. This study is dedicated to her.

ix

TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

ÖZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

CHAPTERS

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Review of Past Studies . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Previous Studies on Nonlinear Static Analysis Procedures 2

1.2.2 Previous Studies on Multi Modal Pushover Analysis Pro-cedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Previous Studies on Generalized Force Vectors . . . . . . 7

1.2.4 Previous Studies on Seismic Analysis of Unsymmetrical-Plan Buildings . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Objective and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 GENERALIZED PUSHOVER ANALYSIS FOR TORSIONALLY COUP-LED SYSTEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Generalized Force Vectors for Torsionally Coupled Systems . . . . 13

2.3 Target Interstory Drift Demand . . . . . . . . . . . . . . . . . . . . 18

2.4 GPA Formulation for Frames in 3D Torsionally Coupled Structures . 18

2.5 Simplified Implementation of GPA in Torsionally Coupled Systems . 24

2.6 Simplified GPA Procedure . . . . . . . . . . . . . . . . . . . . . . 25

x

3 EARTHQUAKE GROUND MOTIONS . . . . . . . . . . . . . . . . . . . . 28

4 CASE STUDY I: EIGHT STORY UNSYMMETRICAL-PLAN BUILDING . 32

4.1 General Information and Modeling of the Building . . . . . . . . . . 32

4.2 Free Vibration Properties . . . . . . . . . . . . . . . . . . . . . . . 36

4.3 Presentations of the Analysis Results . . . . . . . . . . . . . . . . . 36

5 CASE STUDY II: TWENTY STORY UNSYMMETRICAL-PLAN BUIL-DING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 General Information and Modeling of the Building . . . . . . . . . . 72

5.2 Free Vibration Properties . . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Issues Encountered During Analyses . . . . . . . . . . . . . . . . . 78

5.4 Presentation of the Analysis Results . . . . . . . . . . . . . . . . . . 79

6 SUMMARY and CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . 97

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

xi

LIST OF TABLES

TABLES

Table 2.1 tmax values (seconds) for maximum interstory drifts of an eight story tor-

sionally coupled structure composed of four frames in the direction of analysis

(Figure 2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Table 3.1 Selected ground motions and their properties . . . . . . . . . . . . . . . . 29

Table 4.1 Free vibration properties of the eight story structure . . . . . . . . . . . . . 37

Table 5.1 Free vibration properties of the twenty story structure . . . . . . . . . . . . 78

xii

LIST OF FIGURES

FIGURES

Figure 2.1 Effective force vector for second story displacement . . . . . . . . . . . . 14

Figure 2.2 Plan view of the eight story unsymmetrical-plan structure (all units in meters). 19

Figure 2.3 Maximum interstory drifts of the 8-Story structure determined from LRHA

(plan view) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Figure 2.4 Location of Frame k with respect to the center of mass . . . . . . . . . . . 21

Figure 3.1 Acceleration-time histories of the selected ground motions . . . . . . . . . 30

Figure 3.2 Acceleration response spectra of the selected ground motions . . . . . . . 31

Figure 4.1 Elevation view of the frames in the direction of analysis (all units in meters). 33

Figure 4.2 Interstory drift ratios of four frames under GM1 . . . . . . . . . . . . . . 38

Figure 4.3 Mean beam plastic rotations of four frames under GM1 . . . . . . . . . . 38

Figure 4.4 2’nd and 6’th story beam plastic rotations of four frames under GM1 . . . 39

Figure 4.5 Mean column chord rotations of four frames under GM1 . . . . . . . . . . 40

Figure 4.6 2’nd and 6’th story beam end moments of four frames under GM1 . . . . . 41

Figure 4.7 Story shear forces of four frames under GM1 . . . . . . . . . . . . . . . . 42



Figure 4.10 2’nd and 6’th story beam plastic rotations of four frames under GM2 . . . 44






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Figure 5.1 Plan view of the twenty-story unsymmetrical-plan structure (all units in

meters). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Figure 5.2 Elevation view of the frames in the direction of analysis (all units in meters). 74

Figure 5.3 Shear wall modeling approach employed in the case study. . . . . . . . . . 76

Figure 5.4 Interstory drift ratios of five frames under GM1 . . . . . . . . . . . . . . . 81

Figure 5.5 Mean beam plastic rotations of five frames under GM1 . . . . . . . . . . . 82

Figure 5.6 Mean column chord rotations of five frames under GM1 . . . . . . . . . . 83

Figure 5.7 Y-axis shear wall chord rotations under GM1 . . . . . . . . . . . . . . . . 84













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CHAPTER 1

INTRODUCTION

1.1 Statement of the Problem

There are various analysis procedures for the estimation of seismic demands of structures

available in modern engineering practice. They can be mainly grouped as dynamic and static

analysis procedures. Both dynamic and static analysis procedures are further grouped as

linear and nonlinear analysis methods. These procedures vary in performance and computa-

tional effort. Nonlinear dynamic (nonlinear response history) analysis procedure is generally

accepted as the most accurate among the aforementioned analysis methods for predicting

seismic structural response. Although it is accurate, there are several drawbacks which ac-

company this high accuracy. First, its implementation is much more complex than the other

procedures. Second, convergence and stability problems may occur frequently which prevents

achieving correct estimations. Finally, the required amount of computational effort in terms

of run time, computing power and post processing complexity is quite high. Due to these

drawbacks, approximate analysis procedures that yield sufficiently accurate results are usually

preferred in engineering practice. Many of these approximate analysis procedures have also

several shortcomings. For instance, linear and nonlinear static analysis methods mainly em-

ploy statistical modal combination rules in the estimation of actual dynamic response. These

combination rules introduce errors in the obtained results since directional combination of

the modal responses during the actual dynamic response is ignored in statistical combination.

Errors are more prominent in the estimation of member forces since quadratic combination of

modal contributions produces results that are higher than the actual capacities of these mem-

bers. Moreover, many of the proposed nonlinear static analysis procedures are adaptive, that

is, they require an eigenvalue analysis at each static loading increment. As a result, these

1

procedures require developing special computer codes in their implementation. Therefore, a

non-adaptive nonlinear static analysis procedure that includes modal contribution of all modes

without employing statistical combination, and that is conceptually simple and easy to imple-

ment by available structural analysis software may serve as a convenient approach for seismic

demand estimation.

1.2 Review of Past Studies

Generalized pushover analysis procedure that is proposed in this thesis study is a nonlinear

static analysis method. Hence, review of past studies will focus on such analysis procedures.

The review is presented in three sections. In the first section, studies on nonlinear static anal-

ysis procedures are presented. In the second section, studies on more specialized forms of

nonlinear static analysis, namely multi modal pushover analysis procedures are presented.

Third section of the review presents brief information about the past studies completed on the

generalized force vectors, which is a starting point for this thesis study. Since performance

of generalized pushover analysis is evaluated on an unsymmetrical-plan building in this the-

sis study, final section of the review is dedicated to the past studies on seismic analysis of

unsymmetrical plan, or torsionally irregular buildings.

1.2.1 Previous Studies on Nonlinear Static Analysis Procedures

The idea of conducting a nonlinear pushover analysis in order to determine the force defor-

mation characteristics of an equivalent single degree of freedom (SDOF) model was first in-

troduced by Saiidi and Sözen (1981). In the Q-model they have proposed, force-deformation

relationship of the SDOF model is determined by recording the top story displacement and

corresponding overturning moment of a system under a triangular-shaped increasing mono-

tonic loading. Q-model served as a basis for the N2 method proposed by Fajfar and Fischinger

(1988). In the first stage of the method, capacity curve of the multi degree of freedom (MDOF)

system is determined by nonlinear static analysis under a monotonically increasing lateral

force vector. Then, the equivalent SDOF model representing the MDOF system is constructed

by using the capacity curve determined. In the next stage, displacement demand of the SDOF

model is calculated either by employing nonlinear response history analysis of the SDOF sys-

2

tem, or by using an inelastic displacement spectrum. In the final stage, calculated inelastic

demand is converted to the top story displacement of the MDOF system and entire structural

response is obtained from the pushover analysis at the calculated top story displacement. In

order to adapt the lateral force vector to the nonlinear behavior of the MDOF system, Fajfar

and Gaspersic (1996) proposed a force vector proportional to the multiplication of an assumed

displacement shape, namely a post yield displacement shape, with the mass matrix.

Implementation of inelastic demand estimation by using the nonlinear static analysis proce-

dure as a seismic assessment method is also implemented in seismic codes (FEMA-356, 2000;

ATC-40, 1996) . In order to improve the procedure, different methods were proposed for the

determination of displacement demand in a SDOF system. Capacity Spectrum Method pro-

posed by Freeman et al. (1975) employs an equivalent linearization procedure for SDOF de-

mand determination; while displacement coefficient method described in (FEMA-356, 2000)

is based on the idea of multiplying the demand of an elastic SDOF system with conversion

coefficients to calculate the inelastic displacement demand. Considering the advances in the

SDOF demand estimation, Fajfar (2000) proposed a further improvement to N2 Method where

the displacement demands of SDOF systems is determined using the inelastic acceleration-

displacement spectra.

As a seismic performance evaluation method, nonlinear static analysis is discussed in detail by

Krawinkler and Seneviranta (1998). They suggested that unless higher modes do not severely

affect the structure, nonlinear pushover analysis provides a great improvement over elastic

evaluation tools and it can expose design weaknesses of the structures that are not observed

while elastic analysis is employed. Furthermore, estimation of global and local inelastic de-

mands is proved to be sufficient. For inclusion of the effects of the higher modes in the lateral

load vector, studies have also been conducted. Yang and Wang (2000) compared various lat-

eral load distributions and commented that seismic response evaluation with a force vector

constructed by combining the modal forces with square root of the sum of the squares (SRSS)

approach yields better results. Performance of nonlinear static analysis was further evaluated

by Mwafy and Elnashnai (2001) on several reinforced concrete buildings. They stated that

pushover analysis with non-adaptive lateral force distribution produced appropriate results for

low rise and short period buildings and the discrepancies between static and dynamic analysis

results for special and long period buildings are mainly due to the limited capability of the

fixed load distribution in predicting higher mode response in the post-elastic range.

3

As the inability of conventional nonlinear static analysis for capturing the higher mode effects

became apparent with more studies emerging, adaptive pushover analysis procedures were

developed in order to improve the performance of the pushover analysis procedure. An adap-

tive pushover algorithm for an improvement of the capacity spectrum method was proposed

by Bracci et al. (1997). In the algorithm proposed, response of the structure is divided into

four phases, namely, initial elastic state, transition from elastic to inelastic state, formation

of mechanism and failure state. For each phase, separate eigenvalue analysis is conducted

and modal properties are determined, and drift demand curves are constructed using modal

superposition. Gupta and Kunnath (2000) suggested an adaptive spectra-based pushover pro-

cedure. They proposed an algorithm that involves eigenvalue analysis and determination of

modal properties at each step of the lateral load increment. Along with the update of the

modal properties of the structure, spectral demands for each mode are also calculated again

according to this new modal behavior. Modal force vectors with new spectral demands are

constructed for each mode, and nonlinear static analysis is performed independently. Using

the SRSS rule, results of each static analysis is combined and deformations of the structure

for that load increment are obtained. Another adaptive pushover analysis procedure was de-

veloped by Antoniou and Pinho (2004a). They stated that while using combination rules,

equilibrium is not satisfied at each step. Therefore, instead of combining the nonlinear static

analysis results for each mode at the end of each load increment, they proposed combination

of the modal static forces prior to nonlinear static analysis at each step. They stated that a

minor improvement over conventional pushover analysis was achieved. In their companion

paper, Antoniou and Pinho (2004b) developed a new displacement-based adaptive pushover

analysis procedure. A displacement including the contributions from all modes is constructed

through employment of a combination rule and acted upon the structure. They concluded that,

demand estimation improved considerably in comparison to conventional pushover analysis.

Rofooeia et al. (2007) proposed another adaptive pushover analysis algorithm where a spe-

cific load pattern is updated using first few modes. In the formulation of the algorithm, this

update is directly connected to the relative displacement of the roof with respect to the ground.

They concluded that the method has higher accuracy than conventional pushover analysis for

high-rise buildings; however, they state that even this improvement to the pushover analysis

cannot match the results of time history analysis for a high-rise structure.

Nonlinear static analysis procedures become increasingly complex and hard to implement as

4

improvements made. Adaptive pushover analysis is an important enhancement over conven-

tional pushover analysis; however necessity for continuous update of modal properties results

in an increased computational effort. In addition, adaptive or not, conventional nonlinear static

analysis procedures are not all very successful in estimating the effects of higher modes. Ac-

cordingly, new multi modal pushover analysis procedures have been developed over the last

decade.

1.2.2 Previous Studies on Multi Modal Pushover Analysis Procedures

Multi modal pushover analysis is based upon the idea of conducting more than one pushover

analyses with different lateral force distributions which are proportional to the multiplication

of the mass matrix and the mode shape of each different mode. Paret et al. (1996) first sug-

gested this procedure and they stated that after conducting these analyses, capacity curves are

converted to ADRS format (Mahaney 1993). Then, yielding pseudo acceleration values for

the pushover curve of each mode are determined. From the elastic spectrum, elastic pseudo

acceleration values corresponding to the period of each mode are also identified. After these

values are obtained, the Modal Criticality Index (MCI) is calculated for each mode by means

of dividing the elastic pseudo acceleration by yielding pseudo acceleration. The mode which

has the highest MCI value is determined as the critical mode. They concluded that code design

methodologies that are mainly based on conventional pushover analysis do not adequately ad-

dress the higher mode induced story collapses in long-period structures. Sasaki et al. (1998)

later based the Multi-Mode Pushover procedure on the previous MCI determination method.

Using the ADRS format, they obtained the capacity curve of each mode and studied the effect

of higher modes on 17-story and 12-story steel frames. For the 17-story frame, they identified

a high inelastic demand that causes beam hinging in the upper stories. For the 12-story frame,

they also observed significant inelastic demand and accompanying damage. It is pointed out

that for both frames, pushover analysis with triangular load pattern were not successful in the

estimation of these damages.

Modal Pushover Analysis (MPA) is proposed by Chopra and Goel (2002). In the MPA pro-

cedure, pushover analyses are conducted for each mode and modal demands are determined

from response history analyses of inelastic SDOF systems whose capacity curves are con-

structed from the previously performed pushover analyses. Finally, response parameters of the

5

structure at these modal demands are combined by employing SRSS or Complete Quadratic

Combination (CQC) to get the structural response. MPA procedure is based on the assumption

of independent inelastic response for each mode which may not be the case for structures such

as unsymmetrical-plan buildings where coupling of modes is significant. Chintanapakdee and

Chopra (2003) evaluated MPA using regular generic frames and concluded that when a suf-

ficient number of modes are included, drifts determined by MPA is generally similar to the

results from nonlinear response history analysis, while first mode alone pushover procedures

does not display the same accuracy. Also they stated that the deviation from actual response

tends to increase for longer-period frames and larger SDOF-system ductility factors, espe-

cially for taller frames where higher mode distribution are more pronounced. Using vertically

irregular frames, Chintanapakdee and Chopra (2004) also assessed the performance of MPA.

Chopra and Goel (2004) proposed the application of MPA to unsymmetrical-plan buildings.

Upon observing the member forces computed by MPA, Goel and Chopra (2005) stated that

they are unrealistic such that they exceed member capacities. Therefore, they modified the

MPA so that it can estimate the member forces correctly. In their proposed modification, if the

computed member force exceeds the member capacity, it is recomputed from the MPA esti-

mate of member deformations using the member force-deformation relationship. In an effort

to merge the concepts of capacity spectrum method, adaptive pushover analysis procedure

suggested by Gupta and Kunnath (2000) and MPA, Kalkan and Kunnath (2006) proposed a

new pushover analysis procedure. Mao et al. (2008) presented an improved MPA procedure in

which they update the lateral force vector for the first mode yielding point. A new pushover

analysis with the updated force vector is conducted for the first mode; whereas for higher

modes no update is performed. Response is obtained by the modal combination of results

collected from all modes. They concluded that better accuracy is achieved with the improved

MPA procedure. Reyes and Chopra (2011) proposed the three dimensional MPA, where the

analysis algorithm is applied for the two horizontal directions and response obtained is later

combined by the SRSS multi-component combination rule. They also suggested a practical

MPA (PMPA) method which estimates seismic demands directly from the earthquake re-

sponse spectrum rather than response history analysis of the modal SDOF systems for each

ground motion. They stated that the PMPA procedure for nonlinear systems is almost as

accurate as RSA for linear systems.

N2 Method including the effects of higher modes was presented by Kreslin and Fajfar (2011).

6

Taking up the same approach proposed by Fajfar et al. (2005) for asymmetric buildings, their

procedure employs correction factors which are defined as the ratio between the results ob-

tained by elastic modal analysis of the structure and the results obtained by pushover analysis

completed while performing basic N2 method. These correction factors are later used to

determine the local response quantities. They concluded that this extension to N2 method

improved accuracy of the results substantially. They also pointed out that, compared with

the MPA and modified MPA procedures, the extended N2 procedure yielded slightly larger

estimates. These estimates were generally conservative in comparison with the mean results

obtained by nonlinear response history analysis. Recently, Kreslin and Fajfar (2012) com-

bined their previous work on asymmetric buildings in both plan and elevation and extended

the N2 method for these type of structures. The proposed procedure includes using two sets

of correction factors in the extended N2 method, one for displacements and the other one for

story drifts. Correction factor for displacements is obtained from the ratio of the normalized

roof displacements obtained by elastic modal analysis and roof displacements obtained from

pushover analysis. Same procedure is followed for correction factor for story drifts, this time

ratio of story drifts obtained from elastic modal analysis and pushover analysis are calculated.

They concluded that the proposed extended N2 method mostly yields conservative predictions

of higher mode effects.

Jerez and Mebarki (2011) proposed a pseudo-adaptive uncoupled modal response analysis.

They construct an alternative capacity curve for each mode based on the absorbed energy

during pushover analysis. They also state that approximate modal shapes at each pushover

step after yielding can be calculated. Accompanying to these modal shapes, their equivalent

energy-based displacements and corresponding modal participation factors can also be deter-

mined. In their suggested procedure, modal responses obtained using these modal properties

are superimposed rather than employing a modal combination rule and peak responses are

determined.

1.2.3 Previous Studies on Generalized Force Vectors

Generalized force vectors which represent the force vector acting on the system instanta-

neously at the time when a specific response parameter reaches its maximum value were

described by Sucuoglu and Günay (2011). They proposed that these generalized force vectors

7

have the contribution of all modes and if are applied to a structural system, then the maxi-

mum value of the specific response parameter that occurs during dynamic response could be

produced. Selecting the interstory drift as a specific response parameter, these force vectors

were derived. Analytical tests conducted on reinforced concrete frames produced accurate

results in comparison to nonlinear response history analysis. The idea of using generalized

force vectors laid foundation to Generalized Pushover Analysis (GPA) developed in this thesis

study.

Alıcı et al. (2011), further extended the work on generalized force vectors by showing that

using a mean response spectrum for the calculation of force vectors yields similar results to

the mean of the resulting response parameters under a set of strong ground motion excitations.

Hence they have demonstrated the practicality of this approach.

1.2.4 Previous Studies on Seismic Analysis of Unsymmetrical-Plan Buildings

Studies on seismic analysis of unsymmetrical-plan buildings date back to early works on

asymmetric wall-frame structures. Rutenberg and Heidebrecht (1975) proposed an approxi-

mate method for lateral force analysis of asymmetric wall-frame structures. Another approxi-

mate method was suggested by Reinhorn et al. (1977) for the dynamic analysis of torsionally

coupled tall building structures. Previous studies on asymmetric structures were extensively

reviewed by Rutenberg (1992), Rutenberg et al. (1995) and Rutenberg (1998). In his 1998

report, Rutenberg concluded that the studies conducted so far can only be considered as a be-

ginning in understanding the behavior of these structures. Provided that appropriate loading

patterns and eccentricities are selected, he also stated that pushover analysis is a promising

alternative to the linear equivalent lateral force procedure. It is pointed out in his report that

in order to evaluate the approximate methods on asymmetric structures, the problem of de-

termining the displacement target for pushover analysis has to be addressed. Chandler and

Duan (1997) investigated the performance of asymmetric code-designed buildings. Inelastic

earthquake response of single-story asymmetric buildings was worked on by Stathopoulos

and Anagnostopoulos (2003). In their study, performance of simple structures was examined

with the employment of shear-beam type models and plastic hinge idealization of one-story

buildings. Among their conclusions, they deduced that for torsionally flexible unsymmetrical

systems, increase in eccentricity increases ductility demands at the stiff edge and decreases

8

them at the flexible edge. Torsionally unbalanced and irregular concrete buildings have been

the subject of a study conducted by Kosmopoulos and Fardis (2006). According to the provi-

sions of Eurocode 8 (2005), they have developed a computational capability for the analysis,

evaluation and retrofitting of concrete buildings. They verified this computational capability

and modeling approach by comparing the demand estimations for floor displacements and

member damages to the results of pseudo dynamics tests.

In the nonlinear static analysis procedure suggested by Kilar and Fajfar (1997) asymmetric

structures are modeled by using planar macro-elements. For each planar macro-element, a

simple bilinear or multi-linear base shear and top story displacement relationship is assumed

and pushover analysis is conducted. A procedure where higher modal and torsion induced

three-dimensional effects are considered was developed by Moghadam and Tso (1998). In

their procedure, they determined target displacements for resisting elements by conducting

a linear response spectrum analysis of the building. Later, inelastic planar models of these

resisting elements are prepared and they are pushed to these target displacements. Evaluating

the performance of the procedure, they stated that the proposed method leads to good response

estimates for asymmetrical multi-story buildings. A simplified procedure for seismic analysis

of asymmetric plan buildings was proposed by Wilkinson and Thambiratnam (2001) based

on a modification of the shear beam model. According to their conclusions, this modified

shear beam model provides sufficient accuracy and it is applicable to static, quasi-static and

nonlinear dynamic analysis.

Extension of modal pushover analysis to unsymmetrical buildings by Chopra and Goel (2004)

emerged as an important analysis procedure. As difference from 2D case of the procedure,

modal force vectors are composed of lateral forces and torques. When asymmetry is present

in both orthogonal directions, two pushover curves belonging to the structure exist in these

directions. In this case, the authors suggested that pushover curve in the dominant direc-

tion of the mode is to be utilized. After modal pushover analyses with pre-described force

vectors are completed, seismic response of the unsymmetrical-plan structure is obtained by

employing complete quadratic combination (CQC) rule since coupling between modes results

in increased error when SRSS rule is used. The procedure proposed was tested on a mass ec-

centric system for which three different versions were created. MPA is conducted on these

three torsionally stiff, torsionally flexible and torsionally similarly stiff systems and poor per-

formance was observed for the torsionally similarly stiff system due to the strong coupling of

9

modes. Roof displacements were underestimated due to employment of CQC rule.

An extension of the well-known N2 method to asymmetric buildings was proposed by Fajfar

et al. (2005). In this extension, capacity curves are constructed with the N2 method and

seismic demand are determined. Then, torsional effects are determined by a linear modal

analysis of the structure, independently for excitation in two horizontal directions. Results

for the modal analysis are than combined by using SRSS rule. In the final step, structure is

pushed to the demands estimated by N2 method for two horizontal directions and correction

factors to be applied to the results of pushover analyses are determined. These correction

factors are defined as the ratio between the normalized roof displacements obtained by elastic

modal analysis and by pushover analysis. Barros and Almeida (2011) suggested a new multi

modal load pattern based on the relative modal participation of each mode of vibration for

nonlinear static analysis and tested it on two story symmetrical and asymmetric structures.

They defined the modal participation factor they used for determination of the load pattern as

the contribution of each mode to the global response of the system and formed the multi modal

load pattern by multiplying mode shapes by corresponding modal participation factors and

combining them by summation. At the end of their studies, they concluded that while torsional

modes are also significant on the response of structure, the proposed load pattern created much

more accurate results than the conventional load patterns. They also noted that performance of

the proposed load pattern is satisfactory for asymmetric structures; whilst pushover analysis

with a load pattern proportional to the shape of fundamental mode of vibration overestimates

the response of the system.

Upon inspecting the past studies on unsymmetrical-plan buildings, it can be concluded that

there is still much to be determined for the seismic response of these type of structure. Mainly

the methods employed for analysis are generally extended from procedures initially formu-

lated on 2D frames. Therefore, they have some weaknesses in estimating the response of

asymmetric structures on a complete basis.

1.3 Objective and Scope

A generalized pushover analysis procedure which makes use of generalized force vectors

that includes all modal contributions is presented for inelastic seismic response prediction of

10

asymmetric multi degree of freedom three dimensional systems. The procedure is applied

on an eight story unsymmetrical-plan building where significant coupling of modes exist.

The structure is analyzed with various analysis procedures under 10 ground motions. Results

obtained from generalized pushover analysis are compared with nonlinear response history

results. Also included in the comparisons are response predictions from response spectrum

analysis and modal pushover analysis developed by Chopra and Goel (2004). Several re-

sponse parameters such as interstory drift ratios, chord rotations, plastic rotations and member

forces are compared and performance of the procedure is evaluated.

Main objective of the study is to develop a generalized pushover analysis procedure for in-

elastic seismic response prediction of multi degree of freedom three dimensional systems with

asymmetric plan, and test the accuracy of the proposed procedure in predicting the response

parameters.

This thesis is composed of five main chapters. Brief contents are given as follows.

Chapter 1: Statement of the problem and review of past studies on nonlinear

static analysis procedures, multi modal pushover analysis procedures, gener-

alized force vectors and seismic analysis of unsymmetrical plan buildings.

Chapter 2: Detailed explanation and formulation of generalized pushover

analysis procedure.

Chapter 3: Information about ground motions used in analyses.

Chapter 4: Case study of an eight story reinforced concrete unsymmetrical-

plan building with mass asymmetry. Implementation of generalized pushover

analysis and comparison of its results with those of the other seismic demand

determination procedures.

11

Chapter 5: Case study of a twenty story reinforced concrete unsymmetrical-

plan building. Structural system is composed of both moment frames and a

shear wall core offset from center of stiffness. Implementation of generalized

pushover analysis and comparison of its results.

Chapter 6: Summary and conclusions.

12

CHAPTER 2

GENERALIZED PUSHOVER ANALYSIS FOR TORSIONALLY

COUPLED SYSTEMS

2.1 Introduction

Generalized pushover analysis (GPA) for plane frame structures was developed by Sucuoglu

and Günay (2011). GPA is extended to space frames with torsional coupling in this study.

Analytical formulation of generalized pushover analysis for space frames with unsymmetrical

distribution of mass and/or lateral stiffness in plan is presented in this chapter. Definition of

generalized force vectors is given in the first section. The target demand parameters which

are selected as the target interstory drifts and their derivation are explained thoroughly in the

second section. The analysis procedure is outlined in the last section.

2.2 Generalized Force Vectors for Torsionally Coupled Systems

Response parameters achieve their maximum values at different time instants during seismic

response. For a specific response parameter that reaches its maximum value at tmax, there

is an effective force vector acting on the system at that instant (Figure 2.1). This effective

force vector includes contributions from all modes; therefore it is a generalized force vector.

Upon defining the generalized force vector corresponding to a response parameter at tmax and

applying it to the structure, the maximum value of this response parameter can be obtained

by performing an equivalent static analysis. If the system is linear elastic, direct application

of the force vector in a single load step is sufficient for solving the response parameters. For

nonlinear static analysis, however, generalized force vector is applied in an incremental form

13

until a specified target response demand is attained. Interstory drift is selected as the target

response parameter in this study since it gives a good representation of seismic performance

and damage state of the structure at any deformation state. Estimation of the target drift

demand will be discussed in the following sections.

üg

u

feff, 2

(tmax

!!

u2 (tmax)

Figure 2.1: Effective force vector for second story displacement

Generalized force vectors are derived for linear elastic MDOF systems through the application

of modal superposition principle. The maximum value of an arbitrary response parameter can

be obtained at tmax while the system is subjected to an earthquake ground excitation ug(t). The

force vector acting on the system at tmax is defined by the superposition of contributions from

all modes:

f (tmax) =∑

n

fn(tmax) (2.1)

Effective force vector at the n’th mode is given in Equation 2.2 at time tmax:

fn(tmax) = Γn mϕn An(tmax) (2.2)

Parameters in Equation 2.2 are defined below.

Γn = Ln/Mn Ln = ϕTn m l Mn = ϕT

n mϕn (2.3)

14

Here ϕn is the n’th mode shape, m is the mass matrix and l is the influence vector. An(tmax) in

Equation 2.2 can be expressed in terms of the modal displacement Dn at tmax during seismic

response:

An(tmax) = ω2n Dn(tmax) (2.4)

ω2n in Equation 2.4 is the n’th mode vibration frequency, and Dn satisfies the equation of

motion at tmax.

Dn(tmax) + 2 ξn ωn Dn(tmax) + ω2n Dn(tmax) = −ug(tmax) (2.5)

Since Dn(tmax) occurs at a specific time during seismic excitation, it is not possible to deter-

mine it directly from Equation 2.5 if tmax is not known. In the proposed procedure, tmax is

defined as the time when interstory drift (∆ j) of the j’th story reaches its maximum value.

∆ j,max = ∆ j(tmax) (2.6)

The modal expansion of ∆ j(tmax) is given in Equation 2.7:

∆ j(tmax) =∑

n

Γn Dn(tmax) (ϕn, j − ϕn, j−1) (2.7)

Where ϕn, j is the j’th element of the mode shape vector belonging to the n’th mode. Dividing

both sides of Equation 2.7 by ∆ j(tmax) results in the normalized form of this equation:

1 =∑

n

ΓnDn(tmax)∆ j(tmax)

(ϕn, j − ϕn, j−1) (2.8)

The right hand side of Equation 2.8 yields the normalized contribution of each mode n to the

maximum interstory drift of the j’th story at tmax.

While determination of ∆ j(tmax) still depends on tmax, its counterpart in Equation 2.6 can be

estimated through response spectrum analysis (RSA) by employing a modal combination rule.

There are several statistical combination rules available in literature; however since this thesis

15

study focuses on torsionally coupled structures, the selected method should represent this

behavior adequately. Among statistical combination rules, complete quadratic combination

(CQC) is known to be superior to the simpler quadratic combination rule (SRSS) in that

aspect (Wilson et al., 1981). Moreover; even though coupling does not develop in a system,

higher mode effects on seismic response can be evaluated better with CQC compared to SRSS.

Considering these advantages, CQC is chosen as the combination rule employed in RSA.

∆ j,max is expressed in terms of the modal spectral responses obtained with RSA, combined

with CQC in Equation 2.9:

(∆ j,max)2 =∑i=1

∑n=1

ρin[Γi Di (ϕi, j − ϕi, j−1)

] [Γn Dn (ϕn, j − ϕn, j−1)

](2.9)

Here, ρin is the correlation coefficient. Indices i and j denote corresponding modes and ranges

from 1 to N where N is the number of modes. Dn (or Di) is the spectral displacement of the

n’th (or i’th) mode and it is readily available from the displacement response spectrum of

earthquake ground excitation. A normalized form of Equation 2.9 can also be derived by

simply dividing both sides with (∆ j,max)2. The resulting Equation 2.10 shows the normalized

contribution of combined response of i’th and n’th modes to ∆ j,max which is the maximum

interstory drift ratio at the j’th story.

1 =∑i=1

∑n=1

ρin

(Γi

Di

∆ j,max(ϕi, j − ϕi, j−1)

) (Γn

Dn

∆ j,max(ϕn, j − ϕn, j−1)

)(2.10)

In Equations 2.8 and 2.10, normalized contributions of individual modes to the maximum

interstory drift at a specified story are defined from dynamic response history and response

spectrum analyses, respectively. Under the assumption of equality stated in Equation 2.6, the

right-hand sides of Equations 2.8 and 2.10 can be equated:∑n

ΓnDn(tmax)∆ j(tmax)

(ϕn, j−ϕn, j−1) =∑i=1

∑n=1

ρin

(Γi

Di

∆ j,max(ϕi, j − ϕi, j−1)

) (Γn

Dn

∆ j,max(ϕn, j − ϕn, j−1)

)(2.11)

Leaving out similar terms on both sides of Equation 2.11 results in a simplified form:

Dn(tmax) =∑i=1

ρinDn

∆ j,max

[Γi Di (ϕi, j − ϕi, j−1)

](2.12)

16

The terms in brackets in Equation 2.12 is ∆ j,i, or the i’th mode contribution to the maximum

interstory drift at the j’th story determined from RSA. Inserting ∆ j,i for the bracket term in

Equation 2.12 yields a further simplified expression for Dn(tmax):

Dn(tmax) = Dn

∑i=1ρin ∆ j,i

∆ j,max

(2.13)

Equation 2.13 describes Dn(tmax) independent of tmax through RSA accompanied with CQC.

This equality is designated as the modal scaling rule, since modal displacement amplitude of

the n’th mode at tmax is obtained by the multiplication of spectral displacement of this mode

by a scale factor. The modal scaling factor is defined as the ratio of modal contribution to any

response parameter to the maximum of this response parameter calculated at the j’th story.

Interstory drift in the equation is the response parameter selected for this derivation.

An(tmax) can also be determined from Equation 2.13 by multiplying both sides with ω2n:

An(tmax) = An

∑i=1ρin ∆ j,i

∆ j,max

(2.14)

An is the pseudo-spectral acceleration of the n’th mode and obtained from An = ω2n Dn, similar

to Equation 2.4. fn(tmax) can be rewritten via Equation 2.14 in a form that is independent of

time. Substituting An(tmax) from 2.14 into Equation 2.1 yields:

f (tmax) =∑

n

Γn mϕn An

∑i=1ρin ∆ j,i

∆ j,max

(2.15)

As previously stated, formulation presented herein is based on interstory drift at the j’th story.

Consequently, f (tmax) in Equation 2.15 is the generalized force vector that acts on the system

when the j’th story interstory drift reaches its maximum value. Therefore, f (tmax) will be

denoted as f j in the remaining part of the formulation.

Summation for all modes in Equation 2.15 over all terms in the parentheses and regrouping

leads to a final form to the generalized force vector:

17

f j =∑

n

(Γn mϕn An)

∆ j,n

∆ j,max+

N∑i=1i,n

(ρin

∆ j,i

∆ j,max

) (2.16)

The generalized force vector that maximizes the j’th story interstory drift is presented in

Equation 2.16. This is, in fact, the GPA force vector which is applied on the structure in an

incremental form until the target interstory drift demand at the j’th story is obtained. Details of

the analysis procedure are presented in the following sections, starting with the determination

of target interstory drift demand.

2.3 Target Interstory Drift Demand

Maximum interstory drift demand ∆ j(tmax) during ground motion excitation was defined as a

function of Dn(tmax) in Equation 2.7. ∆ j(tmax) can also be expressed with the implementation

of modal scaling rule by substituting Dn(tmax) from Equation 2.13 into 2.7:

∆ j(tmax) =∑

n

Γn (ϕn, j − ϕn, j−1) Dn

∑i=1ρin ∆ j,i

∆ j,max

(2.17)

As discussed previously, Equation 2.17 expresses target drift at the j’th story which reaches

its maximum value when the corresponding f j is acting on the system. Accordingly, ∆ j(tmax)

can be designated as ∆ jt where t stands for target. Regrouping the terms in Equation 2.17

yields the definition for target interstory drift demand at the center of mass:

∆ jt =∑

n

Γn Dn (ϕn, j − ϕn, j−1)

∆ j,n

∆ j,max+

N∑i=1i,n

(ρin

∆ j,i

∆ j,max

) (2.18)

2.4 GPA Formulation for Frames in 3D Torsionally Coupled Structures

Generalized force vectors and their accompanying target interstory drift demands are deter-

mined by using the eigenvectors defined at the diaphragm centers of mass of a 3D structural

system. For a planar 2D frame or a symmetric-plan 3D structure; it is expected that when

18

target interstory drift for a story is reached during nonlinear static analysis, seismic response

of the entire story can be estimated accordingly. In a 2D frame, all structural members are in

the same frame including the center of mass, resulting in consistent deformations and member

forces at the target interstory demand. Similar to the behavior of 2D frames, response of 3D

plan-symmetric structures also shows no variation within a story since torsional effects are

not present. Consequently, deformations and forces of all structural members in a story can

be estimated accurately by employing a single demand control mechanism, namely using one

target interstory demand for each story. However, this is not the case for 3D buildings where

some form of torsional coupling exists.

Figure 2.2: Plan view of the eight story unsymmetrical-plan structure (all units in meters).

19

It is known that due to strong coupling of modes and torsional effects in unsymmetrical-plan

buildings, maximum values of deformations and forces at different frames within a story occur

at different instants, i.e., at different tmax values during dynamic response. Table 2.1 presents

tmax values at each story for the maximum interstory drifts of four frames of the eight story

unsymmetrical plan building for which the plan view is given in Figure 2.2. These results

have been obtained from linear response history analysis (LERHA) under a ground motion

excitation (Superstition Hills, B-PTS315). Each of these frames is part of the eight story

structure where strong coupling of modes occurs due to a mass eccentricity of 15% at each

story. It is clearly seen that tmax varies with frames at each story.

Table 2.1: tmax values (seconds) for maximum interstory drifts of an eight story torsionallycoupled structure composed of four frames in the direction of analysis (Figure 2.2).

Story Frame FE Frame FI Frame SI Frame SE1 5,01 4,98 4,96 4,842 5,06 5,04 5,00 4,963 5,06 5,04 5,03 5,004 5,07 5,06 5,06 5,065 5,16 5,09 5,09 5,096 5,20 5,19 5,14 5,117 5,22 5,19 5,16 5,148 5,24 5,17 5,16 5,15

In addition to this information, Figure 2.3 shows the maximum interstory drifts of frames in

each story that are determined from LERHA under the same ground motion excitation. It

is not possible to estimate this type of behavior displayed in the figure by representing the

response of each story with a simple 2D formulation. In order to reflect this variation in the

individual frame response, generalized force vectors and target interstory drift demands are

defined for the individual frames of a structure at each story instead of employing a single

force vector and target interstory drift demand couple derived at the center of mass. Multi-

ple tmax instances at each story can be “captured” by this approach, yielding better seismic

response estimations.

20

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Insterstory

Drift

(m)

1st Story 2nd Story 3rd Story 4th Story

5th Story 6th Story 7th Story 8th Story

Frame SE Frame SI Frame FI Frame FE

Figure 2.3: Maximum interstory drifts of the 8-Story structure determined from LRHA (planview)

Frame m Frame k

jth Story

!"

CM

!"

#!"

$%"EQ

Figure 2.4: Location of Frame k with respect to the center of mass

21

The n’th mode interstory drifts of Frame k in Figure 2.4 in the direction of ground excitation

is related to the center mass displacements via the distance of frame to the center of mass,

xk along with both translational and rotational components of eigenvectors at the j’th story

(Chopra, 2007). Considering this relation, Equation 2.7 is rewritten as:

∆kj(tmax) =

∑n

Γn Dkn(tmax)

[(ϕny, j − ϕny, j−1

)+ xk

(ϕnθ, j − ϕnθ, j−1

)](2.19)

Dkn(tmax) is the spectral displacement which occurs at the tmax instant for Frame k. Normalized

form of this equation with respect to ∆kj(tmax) is displayed in Equation 2.20:

1 =∑

n

ΓnDk

n(tmax)∆k

j(tmax)


)+ xk


)](2.20)

Similarly, the combined modal response, ∆ j,max which is obtained from RSA through CQC is

defined for Frame k that is shown in Figure 2.4:

(∆kj,max)

2 =∑i=1

∑n=1

ρin

[Γi Di

((ϕiy, j − ϕiy, j−1) + xk (ϕiθ, j − ϕiθ, j−1)

)] [Γn Dn

((ϕny, j − ϕny, j−1) + xk (ϕnθ, j − ϕnθ, j−1)

)](2.21)

Equation 2.21 can also be normalized with respect to (∆ j,max)2:

1 =∑i=1

∑n=1

ρin

ΓiDi

(∆kj,max)

((ϕiy, j − ϕiy, j−1) + xk (ϕiθ, j − ϕiθ, j−1)

) ΓnDn

(∆kj,max)

((ϕny, j − ϕny, j−1) + xk (ϕnθ, j − ϕnθ, j−1)

)(2.22)

The relation defined in Equation 2.6 holds true as in the case of 2D formulation. Taking this

into account, right-hand sides of Equations 2.20 and 2.22 can be equated:

∑n

ΓnDk

n(tmax)∆k

j (tmax)


)+ xk


)]=∑

i=1

∑n=1ρin

[Γi

Di(∆k

j,max)

(ϕiy, j − ϕiy, j−1

)+ xk

(ϕiθ, j − ϕiθ, j−1

)] [Γn

Dn(∆k

j,max)

(ϕny, j − ϕny, j−1

)+ xk


)](2.23)

Normalized form of Equation 2.21 with respect to (∆kj,max)2 is inserted into the right hand side

22

of Equation 2.23 where left hand side represents the normalized form of ∆kj(tmax) in terms of

modal contributions. Leaving out similar terms yields Dkn(tmax) derived for Frame k:

Dkn(tmax) =

∑i=1

ρinDn

∆kj,max

[Γi Di

(ϕiy, j − ϕiy, j−1

)+ xk


)](2.24)

The term in brackets on the right hand side is the i’th mode contribution to the j’th story

interstory drift of Frame k, i.e. ∆kj,i. Then,

Dkn(tmax) = Dn

∑i=1ρin ∆k

j,i

∆kj,max

(2.25)

Here, Equation 2.25 defines Dkn(tmax) in terms of spectral displacement and combined modal

responses. Multiplying both sides by ω2n yields Ak

n(tmax), which, in turn, is used to define the

effective force vector acting at Frame k when the j’th story drift is maximum.

Akn(tmax) = An

∑i=1ρin ∆k

j,i

∆kj,max

(2.26)

Similar to the previously completed derivation, the force vector f k(tmax) acting on Frame k at

the instant when the interstory drift at the j’th story reaches its maximum value is written by

summation of modal contributions. This is achieved by considering the relations defined in

Figure 2.4:

f k(tmax) =∑

n

[Γn mk

(ϕny + xk ϕnθ

)Ak

n(tmax)]

(2.27)

In Equation 2.27, mk is the mass of frame under consideration. In order to obtain the final

form of generalized force vector, Akn(tmax) is inserted from Equation 2.26 into Equation 2.27.

Thus, f kj

for any Frame k is defined as:

f kj =

∑n

Γn mk(ϕny + xk ϕnθ

)Ak

n

∆k

j,n

∆kj,max

+

N∑i=1i,n

ρin

∆kj,i

∆kj,max

(2.28)

23

Equation 2.28 represents the generalized force vector acting at Frame k, when the interstory

drift of the j’th story of that frame reaches its maximum value. In comparison to its 2D

equivalent, Equation 2.16, individual maxima of each frame occurring at different time steps

as in Table 2.1 are taken into account, yielding a more accurate estimation.

Aside from the generalized force vector, the expression for target drift at j’th story of Frame

k is also needed. The modal expansion of the j’th interstory drift occurring at tmax in Frame

k, i.e. ∆kj(tmax) was already defined in Equation 2.19. If Dk

n(tmax) is substituted from Equation

2.25 into Equation 2.19, then ∆kj(tmax) can be expressed in terms of modal spectral responses

by employing the modal scaling rule:

∆kj(tmax) =

∑n

Γn[(ϕny, j − ϕny, j−1

)+ xk


)]Dn

∑i=1ρin ∆k

j,i

∆kj,max

(2.29)

This is in fact the target interstory drift value of the j’th story that is reached when f kj is acting

on Frame k. Similar to 2D formulation, it can be labeled as ∆kjt and expressed in open form in

Equation 2.30:

∆kjt =

∑n

Γn Dn[(ϕiy, j − ϕiy, j−1

)+ xk


)] ∆k

j,n

∆kj,max

+

N∑i=1i,n

ρin

∆kj,i

∆kj,max

(2.30)

Equation 2.30 yields the target interstory drift demand of Frame k at the j’th story under

ground motion excitation. These target interstory drifts are used for demand control at the

k’th frame while conducting pushover analyses with f kj . Therefore f k

j and ∆kjt constitute the

generalized force distribution and the target interstory demand pair for the k’th frame in the

system.

2.5 Simplified Implementation of GPA in Torsionally Coupled Systems

The derivation of two main ingredients of the GPA procedure, GPA force vectors and target

interstory drift demands for 3D torsionally coupled structures have been completed. Building

on these concepts, a simplified implementation of the procedure is introduced in this section.

As discussed in the previous sections, GPA is a nonlinear static analysis procedure relying

24

on multiple pushover analyses conducted by using GPA force vectors derived for each story

of all frames in the system. When the procedure is applied in its complete form, interstory

drift demands at each story and each frame are also defined as target response parameters.

Therefore, the corresponding number of pushovers in GPA is equal to the number of stories

multiplied by number of frames. This, however, creates a significant amount of workload.

For instance, while analyzing an 8-story building with 4 frames in the direction of earthquake

excitation, 32 pushover analyses are needed.

The procedure can be simplified by applying the GPA force vectors at the centers of mass

(Equation 2.16) rather than at each frame individually, while tracking the target interstory

drifts of each frame separately (Equation 2.30). Along a story, different frames reach their

corresponding target interstory drifts at different analysis steps. However, the member re-

sponses obtained at these steps are virtually the same as the ones that are achieved when

performing a full, frame by frame analysis. Application of generalized force vectors defined

at the center of mass significantly reduces the computational effort. To illustrate on the previ-

ously stated 8-story, 4-frame structure, only 8 pushover analyses are performed and 32 target

interstory drift ratios are searched from the frame interstory drifts. Due to the simplicity of-

fered by this approach and the ability to produce results with similar accuracy compared to

frame by frame analysis, the GPA procedure is implemented in this simplified form in the

foregoing presentations.

2.6 Simplified GPA Procedure

The basic approach that the method is built over has been explained in detail above. The

procedure is now developed into an algorithm which can be employed quite easily. At each

pushover analysis conducted, GPA force vector derived for the j’th story of the entire 3D

system (Equation 2.16) is acted upon the structure in an incremental form until the center

of mass reaches the target interstory drift demand calculated according to Equation 2.18. At

each force increment, interstory drifts of all frames at the selected story are determined from

the results recorded during pushover analysis. After they are obtained, target interstory drift

value of each frame is searched. In other words, after the pushover analysis for the j’th story

is performed, interstory drifts for Frame k at each pushover step are browsed in order to find

the step where ∆kjt occurs. When the pushover step is determined, all the deformations and

25

internal forces for structural members of Frame k are acquired from this stage of nonlinear

static analysis. These results represent the response of the structural members when the target

interstory drift, ∆kjt is reached under f j. The global response of the entire structure is estimated

by collecting all member deformations and internal forces at every target interstory drift, ∆kjt

from pushover analyses conducted with each f j and selecting the absolute maximum values

through an envelope algorithm.

Force vectors and target interstory drifts are calculated by using the linear elastic properties of

the structure. It is known that at the advanced stages of pushover analysis, deformations of the

structural system may show quite a different pattern compared to linear elastic behavior due

to nonlinearity. As a result of this phenomenon, the j’th story target interstory drift may not

be achieved occasionally. This situation may sometimes be accompanied with convergence

problems and it may not be possible to achieve the target interstory drift values calculated

by employing linear elastic properties of the structure. In order to overcome this handicap, a

filtering check is performed. After conducting the pushover analysis for the j’th story, it is

checked whether each frame has reached its own ∆kjt, or not. Results of the stories that fail

this check are discarded and the deformations and internal forces of the system are determined

from the envelope results of the pushover analysis of the remaining stories.

To improve the estimation of spectral displacement demand, inelastic spectral displacement,

D∗n may be used for the first two coupled modes in calculating target interstory drifts. In or-

der to determine these first pair of inelastic spectral displacements, modal pushover analyses

of the structure for the first coupled modes are needed. Employing D∗n yields much better

estimation of target interstory drift demand and this in turn increases the chances that conver-

gence problems are not encountered. Therefore, in exchange for a small amount of increased

workload, a good improvement is achieved.

The GPA algorithm which is derived and discussed thoroughly in this chapter is summarized

in the following steps:

1. Modal analysis: Periods (Tn), mode shapes (ϕn) and associated modal properties of the

structure is determined.

2. Response spectrum analysis: Spectral accelerations (An) and displacements (Dn) of

each mode are determined from the corresponding linear elastic spectra. Modal pushover

26

analyses for the first pair of coupled modes can also be conducted in order to estimate

D∗n which is used for estimating the inelastic target interstory drifts. Then, ∆ j,n and

∆ j,max for the center of mass, ∆kj,n and ∆k

j,max for each frame k are obtained by using

these spectral quantities.

3. GPA force vectors: GPA force vectors f j are determined according to Equation 2.16.

4. Target interstory drift demands: Center of mass target interstory drift demands (∆ jt)

which are employed in the incremental nonlinear static analysis under f j are calculated

from Equation 2.18. Frame target interstory drift demands (∆kjt) are also determined by

using Equation 2.30.

5. Nonlinear static analysis: f j are acted upon the system in an incremental form until

∆ jt calculated in step 4 is reached. The convergence check discussed previously is then

employed in order to leave out the stories which have unrealistic responses, if any. Then

the interstory drift record of frame k is searched for ∆kjt. The analysis step that yields

∆kjt is used to compile the member deformations and internal forces at the j’th story

of frame k (target analysis step). This is repeated for every frame where convergence

check has been succeeded.

6. Determination of structural responses: Using the target analysis steps obtained in the

5’th stage of the algorithm, deformations and internal forces of all members at each

frame are determined directly from the deformation state of the structure at the target

interstory drift demand. Entire response of the system is then compiled by employing an

envelope algorithm where absolute maxima of these internal forces and deformations

are selected for every structural member without using any modal combination rule.

These results are listed as final response estimation values.

27

CHAPTER 3

EARTHQUAKE GROUND MOTIONS

The ground motion records which are utilized in the analyses conducted for case studies are

presented in this chapter. Both ordinary and pulse type ground motions are included in the

ground motion set in order to evaluate the performance of Generalized Pushover Analysis

thoroughly under different excitation patterns in case studies. Pulse type ground motions

contain a single significant peak ground acceleration (PGA) value compared to the ordinary

ground motion records. This peak usually occurs in a very short time interval and may stress

the structure considerably during this cycle. In order to estimate the intensity of these pulses,

peak ground velocity values (PGV) are used. Higher PGV usually means a stronger pulse

during the excitation. Ordinary type ground motions, however, lack these distinctive peak

pulses and contain more evenly distributed peaks throughout the record.

Response of the structures to these different types of ground motions may show significant

differences and it is important to consider this behavior. For instance, pulse type ground

motions with a high PGV usually impose a significant demand in a very short time and all

maxima of the member responses are observed almost at this time instant. The instant where

peak demand occurs may result with a substantial amount of yielding and follows with a

change in the modal properties of the structure. Accordingly, the higher mode effects that

are normally anticipated may not be observed. On the other hand, ordinary type ground

motions stress the structure more uniformly during excitation. Since the sudden yielding

accompanying the pulse is absent during the application of the ordinary ground motions, a

better estimation of the individual modal effects can be possible in the analysis.

Eight ground motion records that have been selected in this study are presented in Table 3.1.

All of these records have been downloaded from the PEER strong motion database and used

28

without employing a scaling or any other modification to the original data. For practical

reasons, each ground motion record is given a code number, GM1 to GM8, which will be

used in the remaining part of the study. Second column of the table is the record code that is

associated with the ground motion in the PEER Strong Motion Database. The earthquakes that

these records were produced are given along with the moment magnitude of each earthquake

in the third and fourth columns, respectively. PGA, PGV and peak ground displacement

(PGD) values of each record and type of each ground motion is also shown.

Table 3.1: Selected ground motions and their properties

# GMCODE EARTHQUAKE MwPGA PGV PGD

Type(g) (cm/s) (cm)

GM1 H-E04140 Imperial Valley - 1979 6.5 0.485 37.4 20.1 Pulse

GM2 CLS090 Loma Prieta - 1989 7.0 0.479 45.2 11.3 Pulse

GM3 SPV270 Northridge - 1994 6.7 0.753 84.5 18.7 Pulse

GM4 ORR090 Northridge - 1994 6.7 0.568 51.8 9.0 Ordinary

GM5 ORR360 Northridge - 1994 6.7 0.514 52.0 15.3 Ordinary

GM6 B-PTS315 Superstition Hills - 1987 6.6 0.377 43.9 15.3 Ordinary

GM7 IZT090 Kocaeli - 1999 7.4 0.220 29.8 17.1 Ordinary

GM8 STG000 Loma Prieta - 1989 7.0 0.513 41.2 16.2 Ordinary

Plots of these ground motion records are displayed in Figure 3.1. Acceleration response spec-

tra for 5% damping is shown in Figure 3.2. From the linear elastic spectra, it can be observed

that the ground motions impose different demands on structures with different vibration prop-

erties. In the selection, this property of the ground motions is also considered.

29

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acceleration!(g)

Time!(Seconds)

GM1

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acceleration!(g)

Time!(Seconds)

GM2

0.8

0.4

0.0

0.4

0.8

0 5 10 15 20 25

Acceleration!(g)

Time!(Seconds)

GM3

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acceleration!(g)

Time!(Seconds)

GM4

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acceleration!(g)

Time!(Seconds)

GM5

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20

Acceleration!(g)

Time!(Seconds)

GM6

0.3

0.1

0.1

0.3

0 5 10 15 20 25 30

Acceleration!(g)

Time!(Seconds)

GM7

0.4

0.2

0.0

0.2

0.4

0.6

0 10 20 30 40

Acceleration!(g)

Time!(Seconds)

GM8

Figure 3.1: Acceleration-time histories of the selected ground motions

30

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0.00 0.50 1.00 1.50 2.00 2.50 3.00

Acc

eler

atio

n (

g)

Period (Seconds)

GM1 GM2 GM3 GM4 GM5 GM6 GM7 GM8

Figure 3.2: Acceleration response spectra of the selected ground motions

In the following two chapters, case studies of eight-story and twenty-story unsymmetrical-

plan structures are presented. Results of the analyses performed by using the ground motions

given in this chapter and the performance evaluation of GPA are inspected in detail.

31

CHAPTER 4

CASE STUDY I: EIGHT STORY UNSYMMETRICAL-PLAN

BUILDING

4.1 General Information and Modeling of the Building

The first case study is completed on an eight story unsymmetrical-plan reinforced concrete

building and it is presented in this chapter. Plan of the building was presented in Figure 2.2,

and the elevation view of the frames in the direction of analysis is shown in Figure 4.1. The

building is designed in compliance with the TS-500 and Turkish Earthquake Code (TEC). It

is located in the seismic zone 1 where the local site class is defined as Z3. As required by the

regulations stated in the adopted codes, capacity design principles is employed in design, with

an enhanced ductility level of R=8. Material properties are selected as C25 for concrete and

S420 for reinforcing steel. Uniform member dimensions are selected in the design of beams

and columns. Beams are 30x55 cm2 and columns are 50x50 cm2 throughout the building.

Slab thickness at all floors is 14 cm and live load is 2 kN/m2 according to TS-498. Height

of the stories shows variation only for the first story. Ground story level is 3.5 meters high

and the height of other stories is 3 meters. No basement is considered in the design and the

building is placed directly on fixed supports. Asymmetry and the resulting torsional coupling

are provided by offsetting the mass center of the building from the center of stiffness by 15%

of the plan dimension. Thus, unsymmetrical mass distribution is obtained along the direction

of analysis (Y-axis).

The structure is modeled by using the OpenSees software (2011). Two different models are

prepared for different types of analyses: A linear elastic model is developed for perform-

ing linear elastic response spectrum analysis. A nonlinear structural model is also generated

32

Figure 4.1: Elevation view of the frames in the direction of analysis (all units in meters).

33

in order to perform nonlinear response history analysis, generalized pushover analysis and

modal pushover analysis. In both models, predefined members and material relationships in

the OpenSees platform are utilized. Linear elastic model is composed of “elasticBeamCol-

umn” elements which are defined with geometric properties of the member cross-sections and

shows no material or geometric nonlinearity. In order to account for the cracked section stiff-

ness, gross moments of inertia of the beams are multiplied by 0.5, while that of columns are

multiplied by 0.6.

Due to the complexity and the requirement of high amount of computational effort associ-

ated with the nonlinear analysis, a fully distributed plasticity model is not preferred. Instead,

inelasticity is confined to the member ends. In compliance with this approach, “beamwith-

Hinges” element is utilized in both beams and columns. According to the formulation of this

element, plasticity is “lumped” along a certain hinge length at both ends of the member. Rest

of the element displays linear elastic behavior. These specific hinge lengths are considered as

half of the section depth at all members.

Different plasticity relationships are defined for the aforementioned plasticity zones in beams

and columns. In the case of beams, moment-curvature relationships are defined for each mem-

ber and assigned to the member ends with elasto-plastic hysteresis relations. This approach

is preferred over the more elaborate fiber sections for two major reasons. First, axial loads on

beams show little or no variation during seismic response and remains close to zero. There-

fore a complex section where the moment capacity is updated according to the axial load

variation is unnecessary. Second, when rigid floor diaphragms and fiber sections in beams are

used together, unrealistically high axial loads are observed along the beams. This is a known

issue in OpenSees platform and it is explained by the program authors such that constraining

the neutral axis of a fiber section by a rigid diaphragm results in this type of response.

For each beam cross section, moment-curvature relationships are defined according to the

section detailing. Since demands occurring on each beam along different frames at each story

are different, a single detailing for all beams is not utilized. Instead, beam design is performed

according to the demand on each beam, which resulted in different beam cross-sections for the

structure. This variability in different beam cross-sections is taken into the account while cal-

culating the moment-curvature relationships and hysteresis models are assigned to the beam

ends accordingly.

34

“Hysteretic Material” model is utilized for the hysteresis relationship used in the beams. This

is a bilinear model where moment-curvature relationship is defined. No pinching or deterio-

ration effects are included in the model. For beams, “beamwithHinges” elements are formed

by combining these hysteresis models with the linear elastic properties of the section for the

rest of the member.

Fiber sections are selected in order to introduce plasticity into the columns. The main rea-

son for this choice is that the change in axial load during seismic response is significant in

columns and the bending moment capacity is affected from this change considerably. This

behavior cannot be modeled accurately by defining a moment-curvature relationship obtained

at a single axial load value. Fiber section is a good choice for the described situation since the

response of the section is calculated directly from the material properties and the loads acting

on the member at that instant.

Realistic material models for both reinforcement and concrete are included in the formula-

tion of fiber sections. For reinforcement steel, “Steel01” material model is utilized. It is a

bilinear material model with a very small strain-hardening slope. This behavior is not only

accurate enough, but also relatively simple to represent the reinforcing steel behavior. In the

case of concrete, “Concrete01” model is employed. The model includes a force-deformation

relationship (stress-strain in this case) based on the Modified Kent and Park model (Kent and

Park, 1971) with zero tensile strength. Different material relationships are generated for cover

and core concrete. In the case of core concrete, compressive strength of the confined concrete

(core) is multiplied with Kc which is the amplification factor for confined concrete calculated

according to the aforementioned material model. By selecting such an approach, confinement

effects which occur on the column sections could also be imposed on the analytical model.

Similar to what has been done in the modeling of beams, linear elastic part of the columns are

generated by using the cross-section properties

In the elastic portions of the “beamwithHinges” elements of the nonlinear model, cracked

section stiffness is also considered by following the approach implemented in creating the

linear elastic model. Moment-curvature relationships for the beams and fiber sections for

columns implicitly include this cracking phenomenon and reduction in stiffness; therefore no

extra work is carried out to include these effects in the lumped plasticity regions.

P-Delta effects are also considered in the analytical models. Rather than using linear geomet-

35

ric transformations, P-Delta effects are included in order to obtain a more realistic response.

Rayleigh damping is used in both models where the damping coefficients are obtained from

the 1’st and 3’rd modes. 5% damping is imposed in the solution process for the 1’st and 3’rd

modes.

In order to utilize the described analytical models in various analyses for generating response

results, an automation process is written in MATLAB (R2010b) where response history,

response spectrum, modal pushover and generalized pushover analyses are conducted for the

ground motion set given in Chapter 3. Entire analysis procedures for these methods along with

the pre and post-processing codes are also written in MATLAB. Custom scripts are created

to run OpenSees in conjunction with MATLAB so that the analyses are performed and the

results are produced in a combined procedure.

4.2 Free Vibration Properties

Modal properties of the building are presented in Table 4.1. Among the sixteen torsionally

coupled modes, first nine modes are presented. The first column in Table 4.1 shows the

mode numbers ( X and Y stand for translation dominant mode, and θ stands for the rotation

dominant mode of the Y-θ couple). Corresponding period of each mode is displayed in the

second column. The ordering of modal period indicated that the structure is torsionally stiff.

Effective modal masses (in tons) and the effective modal mass ratios are given in the third and

fourth columns respectively for X and Y directions are presented in the following columns.

4.3 Presentations of the Analysis Results

In the following pages, results compiled from the nonlinear response history analysis (NRHA),

generalized pushover analysis (GPA), modal pushover analysis (MPA) and response history

analysis (RSA) of the eight story structure under the ground motion set consisting of eight

ground motions records described in Chapter 3 are presented. From Figure 4.2 to Figure 4.43,

maximum interstory drift ratios, mean beam plastic rotations, mean column chord rotations,

end moments of beams in the 2’nd and the 6’th stories and story shears of all Y-direction

frames (introduced as FE, FI, SI, SE in Figure 2.2) are plotted. For the first two ground

motions, also presented are the beam plastic rotations of the 2’nd and the 6’th story beams

36

Table 4.1: Free vibration properties of the eight story structure

ModePeriod

(seconds)

EffectiveModal Mass

inY-direction

(M∗n,y) (tons)

EffectiveModal Mass

Ratio inY-direction

EffectiveModal Mass

inX-direction(M∗n,x) (tons)

EffectiveModal Mass

Ratio inX-direction

1X 1.69 0 0 1564 0.8211Y 1.62 1325.28 0.695 0 01θ 1.03 258.81 0.136 0 02Y 0.53 157.47 0.083 0 02X 0.51 0 0 187.91 0.0992θ 0.33 31.21 0.016 0 03Y 0.29 56.69 0.030 0 03X 0.28 0 0 74.22 0.0393θ 0.19 29.58 0.016 0 0

located on these frames.

In general, it is seen that results produced by GPA are satisfactory compared to the NRHA

results which are regarded as the benchmark values. Under moderate to low demand ground

motions, performance of each analysis method is close to each other as can be seen in GM7

and GM8 plots. However, differences increase as seismic demand becomes higher. Despite

increasing differences, GPA mostly yields results sufficiently close to the benchmark values.

GPA is successful in the estimation of higher mode effects especially under GM2, GM3 and

GM4 which excite the second mode. On the other hand, under GM1, GM5 and GM6, some

overshooting in the FE frame results are observed. This is due to the strong demands that

these ground motions impose on the structure which in turn creates high inelastic demands

and increased amounts of nonlinearity in dynamic response. The resulting nonlinearity that

changes the deformation pattern of the structure in dynamic response is difficult to capture

with the static analysis procedures.

For each ground motion, GPA is very successful in terms of estimating the internal forces as

can be seen from the beam end moments and story shears presented in the figures. Since no

modal combination rule is employed in gathering the total response, internal forces can be

estimated almost exactly contrary to the other static analysis methods.

37

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Frame FE, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Frame FI, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Interstory Drift Ratio

Frame SI, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03


Frame SE, GM1

NRHA GPA MPA RSA

Figure 4.2: Interstory drift ratios of four frames under GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Plastic Rotation (rad)

Frame SI, GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM1

NRHA GPA MPA

Figure 4.3: Mean beam plastic rotations of four frames under GM1

38

Second Story Beam Plastic Rotations

Sixth Story Beam Plastic Rotations

0

0,005

0,01

0,015

0,02

0,025

Pla

stic

Ro

tati

on

(ra

d)

Frame FE, GM1

0

0,005

0,01

0,015

0,02

0,025Frame FI, GM1

0

0,005

0,01

0,015

0,02

0,025

Pla

stic

Ro

tati

on

(ra

d)

Frame SI, GM1

0

0,005

0,01

0,015

0,02

0,025Frame SE, GM1

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

Pla

stic

Ro

tati

on

(ra

d)

Frame FE, GM1

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

Frame FI, GM1

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

Pla

stic

Ro

tati

on

(ra

d)

Frame SI, GM1

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007Frame SE, GM1

NRHA GPA MPA

Exterior Beam Exterior Beam Interior

Beam Exterior Beam Exterior Beam Interior

Beam

Exterior Beam Exterior Beam

Interior

Beam Exterior Beam Exterior Beam

Interior

Beam

Figure 4.4: 2’nd and 6’th story beam plastic rotations of four frames under GM1

39

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FE,!GM1

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!FE,!GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FI,!GM1

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!FI,!GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!SI,!GM1

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!SI,!GM1

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Chord!Rotation!(rad)

Frame!SE,!GM1

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01


Frame!SE,!GM1

NRHA GPA MPA RSA

Figure 4.5: Mean column chord rotations of four frames under GM1

40

Second Story Beam End Moments

Sixth Story Beam End Moments

0

100

200

300

400

Moment (kNm)

Frame FE, GM1

0

100

200

300

400

Frame FI, GM1

0

100

200

300

400

Moment (kNm)

Frame SI, GM1

0

100

200

300

400

Frame SE, GM1

0

100

200

300

400

Moment (kNm)

Frame FE, GM1

0

100

200

300

400

Frame FI, GM1

0

100

200

300

400

Moment (kNm)

Frame SI, GM1

0

100

200

300

400

Frame SE, GM1

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam

Figure 4.6: 2’nd and 6’th story beam end moments of four frames under GM1

41

0

1

2

3

4

5

6

7

8

0 150 300 450 600 750 900

Story

#

Frame FE, GM1

0

1

2

3

4

5

6

7

8

0 150 300 450 600 750 900

Frame FI, GM1

0

1

2

3

4

5

6

7

8

0 150 300 450 600 750 900

Story

#

Shear Force (kN)

Frame SI, GM1

0

1

2

3

4

5

6

7

8

0 150 300 450 600 750 900

Shear Force (kN)

Frame SE, GM1

NRHA GPA MPA

Figure 4.7: Story shear forces of four frames under GM1

42

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#

Frame FE, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Frame FI, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#


Frame SI, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02


Frame SE, GM2

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

#

Frame FE, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Frame FI, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

#


Frame SI, GM2

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015


Frame SE, GM2

NRHA GPA MPA


43

Second Story Beam Plastic Rotations

Sixth Story Beam Plastic Rotations

0

0,002

0,004

0,006

0,008

0,01

0,012

Pla

stic

Ro

tati

on

(ra

d)

Frame FE, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012Frame FI, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012

Pla

stic

Ro

tati

on

(ra

d)

Frame SI, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012Frame SE, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012

Pla

stic

Ro

tati

on

(ra

d)

Frame FE, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012Frame FI, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012

Pla

stic

Ro

tati

on

(ra

d)

Frame SI, GM2

0

0,002

0,004

0,006

0,008

0,01

0,012Frame SE, GM2

NRHA GPA MPA

Exterior Beam Exterior Beam Interior

Beam Exterior Beam Exterior Beam Interior

Beam


Interior


Interior

Beam

Figure 4.10: 2’nd and 6’th story beam plastic rotations of four frames under GM2

44

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125

Story

!#

Frame!FE,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075

Frame!FE,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125

Story

!#

Frame!FI,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075

Frame!FI,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125

Story

!#

Frame!SI,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075

Frame!SI,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125

Story

!#


Frame!SE,!GM2

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075


Frame!SE,!GM2

NRHA GPA MPA RSA


45



0

100

200

300

400

Moment (kNm)

Frame FE, GM2

0

100

200

300

400

Frame FI, GM2

0

100

200

300

400

Moment (kNm)

Frame SI, GM2

0

100

200

300

400

Frame SE, GM2

0

100

200

300

400

Moment (kNm)

Frame FE, GM2

0

100

200

300

400

Frame FI, GM2

0

100

200

300

400

Moment (kNm)

Frame SI, GM2

0

100

200

300

400

Frame SE, GM2

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


46

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Frame FE, GM2

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Frame FI, GM2

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Shear Force (kN)

Frame SI, GM2

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Shear Force (kN)

Frame SE, GM2

NRHA GPA MPA


47

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Frame FE, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Frame FI, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#


Frame SI, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03


Frame SE, GM3

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM3

NRHA GPA MPA


48

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!FE,!GM3

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!FE,!GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!FI,!GM3

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!FI,!GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!SI,!GM3

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!SI,!GM3

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#


Frame!SE,!GM3

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015


Frame!SE,!GM3

NRHA GPA MPA RSA


49



0

100

200

300

400

Moment (kNm)

Frame FE, GM3

0

100

200

300

400

Frame FI, GM3

0

100

200

300

400

Moment (kNm)

Frame SI, GM3

0

100

200

300

400

Frame SE, GM3

0

100

200

300

400

Moment (kNm)

Frame FE, GM3

0

100

200

300

400

Frame FI, GM3

0

100

200

300

400

Moment (kNm)

Frame SI, GM3

0

100

200

300

400

Frame SE, GM3

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


50

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Frame FE, GM3

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Frame FI, GM3

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Shear Force (kN)

Frame SI, GM3

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Shear Force (kN)

Frame SE, GM3

NRHA GPA MPA


51

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Frame FI, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#


Frame SI, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03


Frame SE, GM4

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM4

NRHA GPA MPA


52

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!FE,!GM4

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!FE,!GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!FI,!GM4

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!FI,!GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#

Frame!SI,!GM4

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015

Frame!SI,!GM4

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

!#


Frame!SE,!GM4

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01 0,0125 0,015


Frame!SE,!GM4

NRHA GPA MPA RSA


53



0

100

200

300

400

Moment (kNm)

Frame FE, GM4

0

100

200

300

400

Frame FI, GM4

0

100

200

300

400

Moment (kNm)

Frame SI, GM4

0

100

200

300

400

Frame SE, GM4

0

100

200

300

400

Moment (kNm)

Frame FE, GM4

0

100

200

300

400

Frame FI, GM4

0

100

200

300

400

Moment (kNm)

Frame SI, GM4

0

100

200

300

400

Frame SE, GM4

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


54

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Frame FI, GM4

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Shear Force (kN)

Frame SI, GM4

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Shear Force (kN)

Frame SE, GM4

NRHA GPA MPA


55

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Frame FE, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Frame FI, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#


Frame SI, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03


Frame SE, GM5

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM5

NRHA GPA MPA


56

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!FE,!GM5

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!FE,!GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!FI,!GM5

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!FI,!GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!SI,!GM5

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01

Frame!SI,!GM5

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#


Frame!SE,!GM5

0

1

2

3

4

5

6

7

8

0 0,0025 0,005 0,0075 0,01


Frame!SE,!GM5

NRHA GPA MPA RSA


57



0

100

200

300

400

Moment (kNm)

Frame FE, GM5

0

100

200

300

400

Frame FI, GM5

0

100

200

300

400

Moment (kNm)

Frame SI, GM5

0

100

200

300

400

Frame SE, GM5

0

100

200

300

400

Moment (kNm)

Frame FE, GM5

0

100

200

300

400

Frame FI, GM5

0

100

200

300

400

Moment (kNm)

Frame SI, GM5

0

100

200

300

400

Frame SE, GM5

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


58

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Frame FE, GM5

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Frame FI, GM5

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Story

#

Shear Force (kN)

Frame SI, GM5

0

1

2

3

4

5

6

7

8

0 200 400 600 800 1000

Shear Force (kN)

Frame SE, GM5

NRHA GPA MPA


59

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#

Frame FE, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Frame FI, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03

Story

#


Frame SI, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025 0,03


Frame SE, GM6

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM6

NRHA GPA MPA


60

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FE,!GM6

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FE,!GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FI,!GM6

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FI,!GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!SI,!GM6

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!SI,!GM6

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#


Frame!SE,!GM6

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075


Frame!SE,!GM6

NRHA GPA MPA RSA


61



0

100

200

300

400

Moment (kNm)

Frame FE, GM6

0

100

200

300

400

Frame FI, GM6

0

100

200

300

400

Moment (kNm)

Frame SI, GM6

0

100

200

300

400

Frame SE, GM6

0

100

200

300

400

Moment (kNm)

Frame FE, GM6

0

100

200

300

400

Frame FI, GM6

0

100

200

300

400

Moment (kNm)

Frame SI, GM6

0

100

200

300

400

Frame SE, GM6

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


62

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Frame FE, GM6

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Frame FI, GM6

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Shear Force (kN)

Frame SI, GM6

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Shear Force (kN)

Frame SE, GM6

NRHA GPA MPA


63

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM7

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#

Frame FE, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Frame FI, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#


Frame SI, GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02


Frame SE, GM7

NRHA GPA MPA


64

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FE,!GM7

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FE,!GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!FI,!GM7

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FI,!GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#

Frame!SI,!GM7

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!SI,!GM7

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015

Story

!#


Frame!SE,!GM7

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075


Frame!SE,!GM7

NRHA GPA MPA RSA


65



0

100

200

300

400

Moment (kNm)

Frame FE, GM7

0

100

200

300

400

Frame FI, GM7

0

100

200

300

400

Moment (kNm)

Frame SI, GM7

0

100

200

300

400

Frame SE, GM7

0

100

200

300

400

Moment (kNm)

Frame FE, GM7

0

100

200

300

400

Frame FI, GM7

0

100

200

300

400

Moment (kNm)

Frame SI, GM7

0

100

200

300

400

Frame SE, GM7

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


66

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Frame FE, GM7

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Frame FI, GM7

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Shear Force (kN)

Frame SI, GM7

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Shear Force (kN)

Frame SE, GM7

NRHA GPA MPA


67

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#

Frame FE, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Frame FI, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025

Story

#


Frame SI, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02 0,025


Frame SE, GM8

NRHA GPA MPA RSA


0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#

Frame FE, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Frame FI, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

#


Frame SI, GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02


Frame SE, GM8

NRHA GPA MPA


68

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!FE,!GM8

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FE,!GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!FI,!GM8

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!FI,!GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#

Frame!SI,!GM8

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075

Frame!SI,!GM8

0

1

2

3

4

5

6

7

8

0 0,005 0,01 0,015 0,02

Story

!#


Frame!SE,!GM8

0

1

2

3

4

5

6

7

8

0 0,0015 0,003 0,0045 0,006 0,0075


Frame!SE,!GM8

NRHA GPA MPA RSA


69



0

100

200

300

400

Moment (kNm)

Frame FE, GM8

0

100

200

300

400

Frame FI, GM8

0

100

200

300

400

Moment (kNm)

Frame SI, GM8

0

100

200

300

400

Frame SE, GM8

0

100

200

300

400

Moment (kNm)

Frame FE, GM8

0

100

200

300

400

Frame FI, GM8

0

100

200

300

400

Moment (kNm)

Frame SI, GM8

0

100

200

300

400

Frame SE, GM8

NRHA GPA MPA


Interior


Interior

Beam


Interior


Interior

Beam


70

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Frame FE, GM8

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Frame FI, GM8

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Story

#

Shear Force (kN)

Frame SI, GM8

0

1

2

3

4

5

6

7

8

0 200 400 600 800

Shear Force (kN)

Frame SE, GM8

NRHA GPA MPA


71

CHAPTER 5

CASE STUDY II: TWENTY STORY

UNSYMMETRICAL-PLAN BUILDING

5.1 General Information and Modeling of the Building

The second case study is a twenty story unsymmetrical-plan reinforced concrete building. It

is a tall structure where a “C” shaped core is designed to resist most of the lateral forces.

Different from the previous case study where a shift in the mass center has been introduced,

asymmetry in the direction of analysis (y-axis) is imposed by placing the C-core in an offset

position from the mass center. Due to asymmetric nature of the shape of the core and its

position in the plan, stiffness center shifts from the mass center of the building and torsional

coupling develops under horizontal excitation in the y direction.

Plan of the building and the elevation views of the frames in the direction of analysis are

given in Figure 5.1 and Figure 5.2, respectively. TS-500 and TEC are the main guidelines in

the design where capacity design principles are attained similar to the design of the previous

case study. Seismicity and soil conditions assumed in design are seismic zone 1 with local

site class Z3 in TEC. The enhanced ductility level for frame-wall systems (where moment

frames and shear walls are used together to resist the lateral loads) which corresponds to R=7

is intended in the preliminary design. Load reduction factor R has later been revised to 6.87

in the direction of analysis according to TEC, Section 2.5.2.

As defined by the structural layout, all beams have 8 meter spans. These large spans resulted

in a challenging design process where high strength material properties and large section

dimensions has been selected. C50 grade concrete and S420 steel is employed in reinforced

72

Figu

re5.

1:Pl

anvi

ewof

the

twen

ty-s

tory

unsy

mm

etri

cal-

plan

stru

ctur

e(a

llun

itsin

met

ers)

.

73

Figure 5.2: Elevation view of the frames in the direction of analysis (all units in meters).

74

concrete design. Columns are grouped into two as interior and exterior columns, and two

column sections have been designed accordingly. Interior columns are 70x70 cm2 and exterior

columns are 55x55 cm2 throughout the building. Beams are selected as 30x55 cm2 and the

reinforcement design of beams varies among frames and stories according to the demands

calculated. Shear wall thickness is 0.30 meters and each leg of C-core fills an entire span, i.e.

each leg is 8 meters long. Two horizontal legs of the C-core lie on the x-axis (perpendicular to

direction of analysis) and they mainly provide lateral stiffness in that direction. Story height is

4 meters for the first story while it is 3.5 meters for the upper stories. Similar to the previous

case study, there is no basement and the structure is on fixed supports. The slabs have a

uniform thickness of 14 cm. Slab cover is considered as additional dead load with 1.5 kN/m2

intensity. Live load of 2 kN/m2 is also included in the design according to TS-498.

OpenSees software (2011) is employed in the modeling and analysis. In the modeling phase

of the structure, the main challenge exclusive to this case study is the analytical representa-

tion of shear walls. These members are modeled by using frame elements along the shear wall

axis, connected to the joints along the corner axes by rigid beams. The primary advantage of

this approach is its simplicity since it requires less computational effort, both in solution and

post processing. Another alternative that could be considered was employing shell elements

for shear walls. However, this modeling approach has many difficulties that need to be taken

into account. Firstly, shell elements are not used in practice as common as frame elements in

the OpenSees platform and its documentation is insufficient. As a result, verification of shell

models in 3D analysis environment is more difficult than frame element models. Also, com-

putation requirement in terms of effort and runtime accompanying shell elements is another

disadvantage that has to be considered. In the light of these issues, frame members with rigid

beam links are preferred in the modeling of shear walls. The model topology for shear walls

is shown in Figure 5.3.

After deciding on the shear wall model, linear elastic and nonlinear models have been pre-

pared for two different types of analyses. “ElasticBeamColumn” elements are employed for

beams, columns and shear walls in the linear elastic model.

In the nonlinear model, “beamwithHinges” elements are utilized for the beams and columns.

Moment-curvature relationships are defined for beams. Fiber sections with concrete and re-

inforcement material properties are defined for columns. Both of these sectional properties

75

Figure 5.3: Shear wall modeling approach employed in the case study.

are then assigned to the hinge length regions of “beamwithHinges” elements. The detailed

formulation of these sections and elements was discussed in the previous chapter. In the case

of shear walls, “nonlinearBeamColumn” elements are used rather than the “beamwithHinges”

element. Due to the nature of the deformation experienced by shear walls during seismic re-

sponse, plastic hinging is not confined to a certain length at the member ends; therefore an

element formulation with distributed plasticity model is required for accurate response calcu-

lation. As a distributed plasticity element, “nonlinearBeamColumn” members are available

in the OpenSees platform and consequently they are utilized in the modeling. By definition,

“nonlinearBeamColumn” elements require to specify a number of integration points along the

member where the response of the member itself is calculated. Five integration points along

these members are selected.

Shear walls in two directions have different reinforcement designs according to the demands

calculated. Also, along the critical shear wall height of Hcr = 14.5 meters (first four stories),

these designs have been modified according to the regulations stated in TEC. As a result, two

different sections have been defined for shear walls in both directions. The sections are mod-

eled as fiber sections with confined and core concrete properties and steel reinforcements as

in the case of columns, and assigned to the “nonlinearBeamColumn” elements that represent

76

shear walls.

Cracked sections for elastic frame elements are considered similar to what has been done in

the previous case study. In the linear elastic model, gross moments of inertia of beams are

multiplied with 0.4, that of columns are multiplied by 0.6. Gross moments of inertia of shear

walls are multiplied by 0.8. In the nonlinear model, cracked sections are introduced for the

elastic portions of “beamwithHinges” elements, with coefficients of 0.4 for beams and 0.6 for

columns. However, since distributed plasticity members with no elastic portions are used for

the shear walls, no cracked sections are defined for shear walls in the nonlinear model since

fiber sections automatically account for the gravity effects and stiffness of the members are

calculated accordingly.

P-Delta effects are included not only for columns, but also for the shear-walls. 5% damping

is estimated in the analysis where damping coefficients are obtained from the 1’st and 3’rd

modes.

Automation procedures are written in MATLAB (R2010b) where response history, response

spectrum, modal pushover and generalized pushover analyses are conducted for the ground

motion set given in Chapter 3. The general workflow in these written procedures closely fol-

lows the algorithm that is written for the analysis of the 8-Story building discussed in Chapter

4. Pre and post-process scripts are written in MATLAB as well as the codes that are generated

to run the OpenSees within MATLAB environment. Different from the work done in the pre-

vious chapter, 20-story building requires very complex procedures for pre and post-processing

and gathering of results since the number of frame members is large and a considerable large

amount of data is generated in the analyses. As a natural consequence of the increased model

size, runtime for analyses has also increased. Still, an average personal computer is capable

of performing these analyses which emphasizes the importance of adopting simplicity while

constructing the analysis model.

5.2 Free Vibration Properties

Modal properties of the building are presented in Table 5.1. Eigenvalue analysis is performed

for all sixty modes and the nine modes are presented in the table. The first column in Table 5.1

shows the mode number (X and Y stand for translation dominant mode whereas θ stands for

77

rotation dominant mode of the Y-θ couple). Corresponding period of each mode is displayed

in the second column. Effective modal mass (in tons) and the effective modal mass ratio for

X and Y directions are also presented.

Table 5.1: Free vibration properties of the twenty story structure

ModePeriod

(seconds)

EffectiveModal Mass

inY-direction

(M∗n,y) (tons)

EffectiveModal Mass

Ratio inY-direction

EffectiveModal Mass

inY-direction

(M∗n,x) (tons)

EffectiveModal Mass

Ratio inY-direction

1Y 2.36 6725.5 0.559 0 01X 2.10 0 0 7679.1 0.6381θ 1.06 813.98 0.068 0 02Y 0.64 1759.5 0.146 0 02X 0.38 0 0 2283.3 0.1902θ 0.36 133.54 0.011 0 03Y 0.23 870.8 0.072 0 03θ 0.20 316.8 0.026 0 03X 0.15 0 0 153.6 0.013

5.3 Issues Encountered During Analyses

Although simple material models and element formulations have been preferred over more

complex modeling options, the resulting analysis model is still rather complicated compared

with the 8-story structure. Different from the previous case study, presence of shear walls and

their connecting rigid beams in both directions has been one of the main source of modeling

challenge and complexity.

Convergence issues have been encountered in the analysis stage of this case study. Espe-

cially in the nonlinear dynamic time history analysis, the solution failed to converge under

most of the ground motions that have been employed. The poor performance of the model is

considered to depend partly on the inadequate variety of element models and documentation

in the OpenSees platform. For the linear elastic model, this was not the case since “elas-

78

ticBeamColumn” elements represented the behavior accurately enough. However, imposing

the actual shear wall behavior in a nonlinear model with the tools available in the OpenSees

platform had been a real challenge, and consequently the resulting analysis model suffered

from inadequacy in calculating the member forces.

Among the ground motion set discussed in Chapter 3, convergence has been achieved only

for half of the ground motion records. Results for these ground motions are presented in

the following section. Even though the global response of a mixed system (moment frames

and shear wall) is observed in the results (console deformation behavior becomes prominent

as frames gets closer to shear wall), the ground motions selected stressed the structure in a

limited sense.

5.4 Presentation of the Analysis Results

As stated in the previous section, convergence has been achieved under four ground motion

records, GM1, GM2, GM4 and GM5 given in Chapter 3. Results obtained from nonlinear

response history analysis (NRHA), generalized pushover analysis (GPA), modal pushover

analysis (MPA) and response spectrum analysis (RSA) of the twenty story structure are pre-

sented in Figures 5.4 to 5.19. Interstory drift ratios, mean beam plastic rotations, and mean

column chord rotations are plotted for the five frames in the structure in the direction of analy-

sis (denoted as FE, FI, CM, SI and SE). The chord rotations for the shear wall in the direction

of analysis are also plotted.

It is observed from the results that inelastic structural response to the ground motions is very

limited. Matching of GPA with NRHA is very good in GM2 results; it essentially captures the

dynamic response except for SI frame where shear wall is present. In the case of other ground

motion results, close performance of MPA and GPA is notable. This similar performance

could be due to fact that the first mode is dominant; therefore MPA is able to estimate the

fairly straightforward response by simply using the roof displacement. Since no or negligible

higher mode effects are apparent, two methods produce more or less the same results. When

low deformations are encountered, separation between the two may become less observable.

Overall, GPA successfully matches the mean beam plastic chord rotations especially in the

upper stories. Although some overshooting for lower stories is observed at the FE and FI

79

frames under GM5, the errors are minimal. It is also noticeable that the mean column chord

rotation results for all ground motions are closely matched for each analysis methods. GPA

yields the benchmark NRHA values accurately enough for column chord rotations.

Shear wall chord rotations are too small to make a reasonable comparison but GPA seems to

be successful in estimating the deformation trend observed in NRHA.

80

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01

Story

#

Frame FE, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01

Frame FI, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01

Story

#


Frame CM, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01


Frame SI, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01

Story

#


Frame SE, GM1

NRHA GPA MPA RSA

Figure 5.4: Interstory drift ratios of five frames under GM1

81

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#

Frame FE, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Frame FI, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#


Frame CM, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006


Frame SI, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#


Frame SE, GM1

NRHA GPA MPA

Figure 5.5: Mean beam plastic rotations of five frames under GM1

82

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004

Story

!#

Frame!FE,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame!FE,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004

Story

!#

Frame!FI,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame!FI,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004

Story

!#


Frame!CM,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002


Frame!CM,!GM1

NRHA GPA MPA

Figure 5.6: Mean column chord rotations of five frames under GM1

83

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004

Story

!#

Frame!SI,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame!SI,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004

Story

!#


Frame!SE,!GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002


Frame!SE,!GM1

NRHA GPA MPA

Figure 5.6: Mean column chord rotations of five frames under GM1 (continued)

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.00025 0.0005 0.00075 0.001

Story

#

Chord Rotation (rad)

Shear Wall I!End Chord Rotations, GM1

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.00025 0.0005 0.00075 0.001


Shear Wall J!End Chord Rotations, GM1

NRHA GPA MPA

Figure 5.7: Y-axis shear wall chord rotations under GM1

84

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Sto

ry#

Frame FE, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Frame FI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Sto

ry#


Frame CM, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012


Frame SI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Sto

ry#


Frame SE, GM2

NRHA GPA MPA RSA


85

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Sto

ry#

Frame FE, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Frame FI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Sto

ry#


Frame CM, GM2

0

1

2

3

4

5

6

7

8

9

10

11

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13

14

15

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17

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19

20

0 0.002 0.004 0.006


Frame SI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Sto

ry#


Frame SE, GM2

NRHA GPA MPA


86

I-Ends J-Ends

0

1

2

3

4

5

6

7

8

9

10

11

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0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame FE, GM2

0

1

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3

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5

6

7

8

9

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20

0 0.0005 0.001 0.0015 0.002 0.0025

Frame FE, GM2

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0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame FI, GM2

0

1

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3

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5

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7

8

9

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11

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13

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0 0.0005 0.001 0.0015 0.002 0.0025

Frame FI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame CM, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025


Frame CM, GM2

NRHA GPA MPA


87

I-Ends J-Ends

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame SI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Frame SI, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame SE, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025


Frame SE, GM2

NRHA GPA MPA


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015

Sto

ry#


Shear Wall I-End Chord Rotations, GM2

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015


Shear Wall J-End Chord Rotations, GM2

NRHA GPA MPA


88

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008

Sto

ry#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008

Frame FI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008

Sto

ry#


Frame CM, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008


Frame SI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008

Sto

ry#


Frame SE, GM4

NRHA GPA MPA RSA


89

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Frame FI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame CM, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005


Frame SI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame SE, GM4

NRHA GPA MPA


90

I-Ends J-Ends

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame FE, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame FE, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame FI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame FI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame CM, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002


Frame CM, GM4

NRHA GPA MPA


91

I-Ends J-Ends

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#

Frame SI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002

Frame SI, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Sto

ry#


Frame SE, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002


Frame SE, GM4

NRHA GPA MPA


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015

Sto

ry#


Shear Wall I-End Chord Rotations, GM4

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015


Shear Wall J-End Chord Rotations, GM4

NRHA GPA MPA


92

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Story

#

Frame FE, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Frame FI, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Story

#


Frame CM, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012


Frame SI, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006 0.008 0.01 0.012

Story

#


Frame SE, GM5

NRHA GPA MPA RSA


93

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#

Frame FE, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Frame FI, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#


Frame CM, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006


Frame SI, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.002 0.004 0.006

Story

#


Frame SE, GM5

NRHA GPA MPA


94

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Story

!#

Frame!FE,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025

Frame!FE,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Story

!#

Frame!FI,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025

Frame!FI,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Story

!#


Frame!CM,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025


Frame!CM,!GM5

NRHA GPA MPA


95

I Ends! J Ends!

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Story

!#

Frame!SI,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025

Frame!SI,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.001 0.002 0.003 0.004 0.005

Story

!#


Frame!SE,!GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.0005 0.001 0.0015 0.002 0.0025


Frame!SE,!GM5

NRHA GPA MPA


0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.00025 0.0005 0.00075 0.001

Story

#


Shear Wall I!End Chord Rotations, GM5

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

0 0.00025 0.0005 0.00075 0.001


Shear Wall J!End Chord Rotations, GM5

NRHA GPA MPA


96

CHAPTER 6

SUMMARY and CONCLUSIONS

6.1 Summary

Generalized pushover analysis is developed in this study as a nonlinear static analysis proce-

dure for estimating the inelastic response of torsionally coupled structures under earthquake

ground motions. The analysis procedure is built upon the generalized force vector concept

which is defined as the effective force vector acting on the system at the instant tmax when

a specific response parameter reaches its maximum value. The specific response parameter

is selected as the interstory drift of j’th story (∆ j). Although this generalized force vector is

dependent on tmax in a true dynamic response history analysis, it is approximately defined in

terms of modal spectral responses obtained via response spectrum analysis in this study by

using a modal scaling rule. The rule is applied for interstory drift contribution of all modes

to the j’th interstory drift (∆i, j), which yields the generalized force vector in terms of modal

spectral responses. They are applied to the structure in an incremental form until the target

response parameter is attained. Target response parameter, which is the interstory drift, is

also calculated for all stories by using the same modal scaling rule. By considering the nature

of the response of unsymmetrical plan structures, this formulation is further extended to a

frame-by-frame approach where different target drift demands and generalized force vectors

are determined for each frame at each story. It is observed that in an unsymmetrical-plan

building, different frames on the same story attain their tmax values at different time instants.

Therefore the aforementioned approach which accounts for this phenomenon is expected to

provide better response estimation. A practical implementation of the procedure is also dis-

cussed where the structure is pushed with generalized force vectors defined at the center of

mass meanwhile the interstory drift demands for the individual frames are employed as tar-

97

get demand parameters. This simplification requires less computational effort and runtime

while providing the results with the same accuracy. At the final stage of the analysis, results

gathered for each frame from the pushover analyses under all generalized force vectors are

combined in an envelope algorithm.

Two case studies are prepared in order to evaluate the performance of the proposed analysis

procedure. The first one is an eight story moment frame structure with mass eccentricity

while the second one is a twenty story frame-wall building where the shear wall core is placed

with offset from the center of mass. Both of the structures are analyzed under eight ground

motions. Nonlinear response history, generalized pushover, modal pushover and response

spectrum analyses are performed in both case studies. Interstory drift ratios, mean beam

plastic rotations and mean column chord rotations of each frame at each story obtained from

different analysis procedures are compared. Story shear forces, second and sixth story beam

end moments of each frames are also compared for the eight story structure.

6.2 Conclusions

According to the results obtained in this study, the following conclusions are reached:

• For the majority of the ground motion records, GPA is successful in estimating the

response of eight story structure compared to benchmark NRHA. Higher mode effects

are well captured and this is best observed in interstory drift ratios and mean beam

plastic rotations. It is also noted that if matching of RSA and LERHA (Linear Elastic

Response History Analysis) is good enough, then GPA yields very good estimates since

GPA includes RSA as a prerequisite. Using the inelastic spectral displacement demand

D∗n for the first mode couple results in an improvement for some of the ground motions.

• GPA is very accurate in estimating the internal forces. No modal combination is em-

ployed while compiling the results at the final step of GPA, therefore it yields the forces

directly as they were recorded during analysis. Both of the case studies are designed

with capacity design principles and this is observed at the beam end moment compar-

isons. NRHA and GPA give almost the same moment values for most of the cases

since capacity of the beam member is reached in both analyses. However, other anal-

ysis methods that employ modal combination rule overestimate the force values and

98

exceed member capacities, which is unrealistic. Story shear forces estimated by GPA

also match the NRHA results well, which underlines the success in force estimation.

• Modeling challenges that is encountered in the twenty story frame-wall structure made

the analysis difficult. For the four ground motions where convergence had been achieved,

member deformations were small. The massive shear wall core gives a high lateral stiff-

ness to the structure and the global response stayed at minimum. Also torsionally stiff

behavior of the structure did not result in prominent higher mode effects. Although the

recorded deformations were limited, GPA was successful in estimating the response

under ground motions where the higher mode effects were significant. However shear

wall rotations were very small and it was difficult to make an evaluation of the proposed

procedure based on these values. In general, it can be suggested that GPA has the poten-

tial to analyze complex high-rise structures which are located on the long period range

of the response spectrum and GPA is successful in capturing the global deformation

trend of the system. However, this needs to be evaluated with analysis models that are

prepared on a more sound software environment that has the necessary tools to model

these complex structures.

• Lack of convenient material models and element formulations for 3D modeling of shear

walls is an issue in the OpenSees platform. Due to this limitation, approximate model-

ing solutions that are employed may not reflect the actual behavior properly as discussed

above. Also issues with rigid diaphragms and fiber sections which result in unrealisti-

cally high axial loads in beams when fiber sections are used is another drawback of the

software. These problems should be further investigated and taken into consideration

while modeling with OpenSees.

• The potential of GPA as an approximate analysis method is noticeable when the analysis

results are investigated. It is a good alternative to the nonlinear response history analysis

and its estimations for unsymmetrical-plan structures are generally acceptable. For long

period systems, its accuracy may further be tested in different analysis environments

that are capable of representing the behavior of structural members like shear walls

successfully in a 3D model.

99

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GENERALIZED PUSHOVER ANALYSIS FOR UNSYMMETRICAL …